i
E olu ion o Coope a ion in Mul ilaye Ne wo ks
Paulo Ba e o Vale iano de Albuque que Sa dinha
Disse a ion p esen ed as pa ial equi emen o ob aining
he Mas e ’s deg ee in In o ma ion Managemen
i
Managemen School
Ins i u o Supe io de Es a ís ica e Ges ão de In o mação
Uni e sidade No a de Lisboa
EVOLUTION OF COOPERATION IN MULTILAYER SOCIAL
NETWORKS
by
Paulo Ba e o Vale iano de Albuque que Sa dinha
Disse a ion p esen ed as pa ial equi emen o ob aining he Mas e ’s deg ee in In o ma ion
Managemen , wi h a specializa ion in In o ma ion Sys ems and Technologies Managemen
Ad iso : p o . Dou o Flá io L. Pinhei o (NOVA IMS)
Co Ad iso : p o . Dou o F ancisco C. San os (INESC-ID/Ins i u o Supe io Técnico)
July 2020
ii
To my pa en s, Eulália and F ancisco,
wi h lo e and g a i ude
iii
ACKNOWLEDGEMENTS
I would like o hank my supe iso , D Flá io Pinhei o, o his op imism, o he
oppo uni y c ea ed and o p o iding me guidance and eedback h oughou his p ojec .
Wi hou his suppo I would no ha e been able o comple e his esea ch.
I would also like o hank my co-supe iso D F ancisco San os o his insigh ul
commen s on his esea ch.
The pape s I had he chance o ead om hem we e inspi ing and decisi e in my
decision o unde ake his esea ch.
I would also like o hank Ma ga ida o he cons an encou agemen , in e es and help
in he subjec s I s udied du ing he mas e 's cu icula componen leading o his hesis.
Las bu no leas I wan o hank my amily and iends o hei incen i e and o
pu ing up wi h he ime I in es ed in his mas e p og am dep i ing me o social
ga he ings.
i
ABSTRACT
Indi iduals ake pa in mul iple laye s o ne wo ks o in e ac ions simul aneously.
These in e dependen ne wo ks accoun o he di e en so o social ies indi iduals
main ain pe laye . In each laye indi iduals pa icipa e in N-Playe Public Goods Games
whe e bene i s collec ed inc ease wi h amoun s in es ed. I is, howe e , emp ing o be a
ee- ide , i.e., o ake ad an age o he common pool wi hou con ibu ing o i , a si ua ion
om which a social dilemma esul s. This hesis o e s new insigh s on how coope a ion
dynamics is shaped by mul iple laye s o social in e ac ions and di e si y o con ibu ions
in es ed pe game. To his end, we eso o E olu iona y Game Theo y and Ne wo k
Science o p o ide a con enien amewo k o add ess he mos impo an p o o ypical
social con lic s and/o dilemmas in la ge ne wo ked popula ions. In pa icula , we
p opose a no el mean- ield app oach capable o acking he sel -o ganiza ion o
Coope a o s when co-e ol ing wi h De ec o s in a mul ilaye en i onmen . We show ha
he eme ging collec i e dynamics, which depends (i) on he unde lying laye ne wo ks o
in e ac ions and (ii) on he c i e ia o sha e a ini e in es men ac oss all games, o en does
no bea any esemblance wi h he local p ocesses suppo ing hem. Ou indings sugges
ha , whene e indi idual in es men s a e dis ibu ed among games o laye s, esilience
o coope a ion agains ee- ide s inc eases wi h he numbe o laye s, and ha
coope a ion eme ges om a non- i ial o ganiza ion o coope a ion ac oss he laye s. In
opposi ion, unde cons an , non-dis ibu ed in es men s, he le el o coope a ion shows
li le sensibili y o a ia ions in he numbe o laye s. These indings pu in e idence he
impo ance o asymme ic con ibu ions ac oss games and social con ex s in he
eme gence o human coope a ion.
KEYWORDS
E olu iona y Game Theo y, Ne wo ks, E olu ion, Coope a ion, Public Goods Games,
Mul ilaye , A e age G adien o Selec ion

INDEX
1. In oduc ion ................................................................................................................. 1
1.1. Thesis S uc u e ..................................................................................................... 4
2. Li e a u e Re iew ......................................................................................................... 7
2.1. The P oblem O Coope a ion ................................................................................ 7
2.1.1. Two Pe son Games, he P isone ’s Dilemma .............................................. 8
2.1.2. Public Goods Games .................................................................................. 11
2.1.3. E olu iona y dynamics in ini e popula ions ............................................. 13
2.1.4. The Replica o Dynamics .......................................................................... 22
2.2. The Mechanisms o Coope a ion......................................................................... 26
2.3. The Science o Ne wo ks..................................................................................... 30
2.3.1. Models o Ne wo ks ................................................................................... 35
2.3.1.1. Random Ne wo ks .......................................................................... 35
2.3.1.2. Ho and Ne wo ks ............................................................................ 38
2.3.1.3. Scale-F ee Ne wo ks ....................................................................... 38
2.3.1.4. Mul ilaye s ...................................................................................... 40
2.3.1.5. De ini ion o a Mul ilaye Ne wo k ................................................ 42
2.3.1.6. Deg ee-Deg ee Co ela ion ............................................................. 44
2.3.1.7. O e lapping .................................................................................... 46
2.4. E olu iona y Games on S uc u ed Popula ions.................................................. 46
2.4.1. Accumula ed e sus A e age Payo s ....................................................... 48
2.4.2. Upda e Rules .............................................................................................. 49
2.4.3. G adien o Selec ion ................................................................................. 52
2.4.4. The S uc u e o Social G aphs .................................................................. 55
2.5. O he Rele an Bibliog aphy o he Thesis ........................................................ 57
2.5.1. Mul ilaye ne wo ks ................................................................................... 58
2.5.2. Coope a ion in Mul ilaye Ne wo ks ......................................................... 58
3. Model And Me hods ................................................................................................... 63
3.1. Model ................................................................................................................... 63
3.2. Compu e Simula ions ......................................................................................... 65
3.2.1. Nume ical Me hods .................................................................................... 70
4. Resul s and Discussion ............................................................................................... 72
4.1. Topological Ensla emen unde Dis ibu ed In es men s and La ge
Numbe o Laye s ............................................................................................... 77
i
4.2. Laye and Popula ion Pola iza ion ...................................................................... 80
4.3. Agg ega ed A e age G adien o Selec ion ......................................................... 85
5. Conclusions ............................................................................................................... 91
5.1. Fu u e Wo k......................................................................................................... 93
6. Bibliog aphy ............................................................................................................... 95
Appendix A. Algo i hm o Payo and AGoS Calcula ion ...................................... 102
Appendix B. Deg ee-Deg ee Co ela ion and O e lapping ...................................... 107
Appendix C. C i e ia and Topological Ensla emen ................................................ 114
Appendix D. Theo e ical Agg ega ed G adien o Selec ion wi h
Dis ibu ed In es men ........................................................................ 129
Appendix E. Mul ilaye s wi h di e en Types o Ne wo ks .................................... 133
ii
LIST OF FIGURES
Figu e 1- T-S Quad an s o 2-Playe Games ............................................................................ 9
Figu e 2- Ma ko Chain o a Laye wi h S a e e lec ing he Numbe o Coope a o s .......... 16
Figu e 3- Examples o supe exponen ial and subexponen ial g ow hs. ................................... 25
Figu e 4- Mechanisms o Coope a ion. .................................................................................... 27
Figu e 5- Ne wo k Adjacency Ma ix ...................................................................................... 31
Figu e 6- Examples o Ne wo k Asso a i i y and Disasso a i i y. ....................................... 33
Figu e 7- Examples o Clus e ing Coe icien Calcula ions o 3 Ne wo ks. .......................... 34
Figu e 8- F om Regula o Random Newo ks. ......................................................................... 37
Figu e 9- Poisson e sus Powe -law Dis ibu ions. ................................................................. 39
Figu e 10- Mul ilaye Ne wo ks. .............................................................................................. 43
Figu e 11-E ec s o Popula ion S uc u e on 2-Playe Games. ............................................... 48
Figu e 12- Applica ion o Upda e Rule in a Mul iplex ............................................................ 52
Figu e 13-A e age G adien o Selec ion ................................................................................ 55
Figu e 14- Le el o Coope a ion as a Func ion o Ne wo k ype, In es men C i e ia,
Numbe o Laye s, In ensi y o Selec ion (β) and Enhancemen Fac o (F). ........ 72
Figu e 15- Quasi-S a iona y P obabili y o an 8-Laye Mul ilaye wi h BA Ne wo ks
and Baseline In es men C i e ia .......................................................................... 74
Figu e 16- Le el o Coope a ion in Mul ilaye Ne wo ks in 2-Playe Dis ibu ed
P isone Dilemma (=0.1). .................................................................................... 75
Figu e 17- Expe imen al Quasi-S a iona y Dis ibu ion o Mul ilaye s a e aged
ac oss Ne wo k Laye s. ......................................................................................... 78
Figu e 18- Indi idual S a egy Consis ency ac oss an 8 laye BA mul ilaye wi h
k=4,β=0.1,N=1000. .................................................................................... 81
Figu e 19- Sa u a ion in Mul ilaye s as he Numbe o Laye s a y. ..................................... 82
Figu e 20- Sa u a ion o Laye s in a Mul ilaye as In es men C i e ia and he Numbe o
Laye s a y. ........................................................................................................... 83
Figu e 21- Laye and Node Consis ency in a Mul ilaye as In es men C i e ia and he
Numbe o Laye s a y. ......................................................................................... 84
Figu e 22- A e aged Agg ega ed G adien o Selec ion (AGoS) a e aged ac oss
Laye s and Time. ................................................................................................... 85
Figu e 23- Agg ega ed G adien o Selec ion (AGoS) o 4 Laye s Mul ilaye o e Time. .... 87
Figu e 24- Agg ega ed G adien o Selec ion (AGoS) o Mul ilaye s a e aged o e Time. . 88
Figu e B-1- Example o a Deg ee-Deg ee Co ela ion Ma ix ............................................. 107
Figu e B-2- A e age Le el o Coope a ion o Deg ee-Deg ee Co ela ion e sus
In ensi y o Selec ion (β). ................................................................................. 109
iii
Figu e B-3- A e age Le el o Coope a ion o Deg ee-Deg ee Co ela ion e sus
Enhancemen Fac o . ......................................................................................... 111
Figu e B-4- A e age Le el o Coope a ion in BA Mul ilaye s as a unc ion o Link
O e lapping and Enhancemen Fac o . ............................................................. 112
Figu e C-1- Payo s Dis ibu ion o Ho and Mul ilaye wi h In es men pe Game
e sus Numbe o Laye s. ................................................................................. 117
Figu e C-2- Time Se ies o E olu ion o Coope a ion Le el in 16 Laye Ho and
Mul ilaye wi h In es men dis ibu ed pe Game. ........................................... 118
Figu e C-3- Ma ko Chain co esponding o D unka d’s Walk P ocess .............................. 119
Figu e C-4- Theo e ical esul s o a Mean-Field app oxima ion o a Laye wi h
Nodes wi h equal Payo . .................................................................................. 121
Figu e C-5- Topological and C i e ia Ensla emen in Mul ilaye s wi h 16 Laye s,
In es men dis ibu ed pe Game. ..................................................................... 123
Figu e C-6- Final Le el o Coope a ion as a Func ion o ini ial One. .................................. 124
Figu e C-7- Topological and C i e ia Ensla emen o 16-laye s Mul ilaye , 1000
nodes (N) pe laye . ........................................................................................... 125
Figu e C-8 - Domain o Topological Ensla emen o In es men dis ibu ed pe Game. .... 126
Figu e C-9 - Consis ency T end in Mul ilaye s wi h In es men dis ibu ed pe Game. ....... 127
Figu e C-10- Consis ency in Ho and Mul ilaye s as he Numbe o Laye s a ies. .............. 127
Figu e D-1- Theo e ical AGoS o Ho and Ne wo ks in Mul ilaye s wi h In es men
pe Game C i e ia .............................................................................................. 132
Figu e E-1- Mul ilaye wi h 8 laye s BA o Ho and s. Mixed Mul ilaye wi h 4 laye s
Ho and plus 4 laye s BA. .................................................................................. 133
5
a ou ed. Ye , coope a ion is obse ed on many le els o biological and social o ganiza ions.
The nex sec ion a emp s o explain mechanisms ha may be pu in place o na u al selec ion
o a ou coope a ion and how hey o se he cos s o coope a ion by causing Coope a o s o
also be on he ecei ing end mo e o en. An o e iew o he mos ele an concep s om he
no el ield o Ne wo k Science o be used h oughou he ex ollows wi h a cha ac e iza ion
o he ne wo ks ha suppo he expe imen a ions. Game dynamics, whe he un olding o e
con inuous o disc e e ime, go e ned a he mic oscopic-agen le el by s ochas ic ules o
ollowing de e minis ic ules modelling he di ec ion and in ensi y o selec ion as a unc ion
o ela i e popula ion concen a ions is ocused nex . Fo he sake o comple eness, chap e 2
inishes wi h a su ey on publica ions ela ed o he opics pu sued in his hesis.
Expe imen s we e conduc ed ia compu e simula ions ha un on a amewo k buil ad
hoc, implemen ing mul iple ins ances o agen s wi h beha iou s go e ned by s ochas ic
p ocesses. Chap e 3 aims o explain his amewo k, o de ail he models cons uc ed, o
discuss he implemen a ion op ions a ailable and he a ional beyond he op ions aken.
Independen pa ame e s, bo h opological and beha iou al, and me ics o collec (ou pu s) as
well as he me hodologies ollowed a e iden i ied.
Chap e 4 p esen s he main esul s collec ed om he simula ions, highligh ing he
indi idual in luence o he pa ame e s on he coope a ion le els a ained by he sys em. In
pa icula , i shows how he combina ion o opology and he c i e ia o deciding on how o
sha e a ini e in es men among all games each indi idual pa icipa es in impac s he
coope a ion le el achie ed, leading, in ex eme ci cums ances, o an ensla emen in which
he le el o coope a ion a ained becomes insensi i e o o he en i onmen pa ame e s. A
limi ed se o pa ame e s dic a e he ensla emen condi ion. When his condi ion se s in,
changes in o he pa ame e s ha e a ma ginal e ec on he le els o coope a ion a ained which
is p ese ed since mul ilaye ini ializa ion ime. Ma hema ical explana ions o ensla emen
o occu and o i s consequences a e explo ed. In o de o be e unde s and he dynamics
unleashed on complex sys ems o en un ela ed o he s ochas ic ules p og ammed a agen -
le el he o me a e buil upon, one has eso ed o A e age G adien o Selec ion (AGoS), a
ime and con ex independen me ic, because o being a e aged ac oss ime and he
popula ion, bu dependen on pa ame e s such as he ne wo ks suppo ing he in e ac ions
among indi iduals ha e lec s a end o he e olu ion o he numbe o Coope a o s o e
ime. The use o AGoS ini ially concei ed o a single laye is gene alized o a mul ilaye case.
AGoS esul s a e in e p e ed and co ela ed wi h ensla emen . Finally, in chap e 5, we d aw
concluding ema ks and discuss u u e s eps.

6
The hesis includes a numbe o appendixes complemen ing he esul s shown in chap e
4. Appendix A complemen s chap e 3 in p esen ing he algo i hm concei ed in o de o
minimize he du a ion o he simula ions execu ed. In appendix B, we discuss he e ec s o
deg ee-deg ee co ela ion and o e lapping on he e olu ion o coope a ion le els eached by
he mul ilaye a e explo ed. In appendix C, a ma hema ical analysis explaining he opological
and in es men c i e ia ensla emen is de eloped. Wi h condi ions o ensla emen me and
aking a mean ield app oach, expec ed AGoS is an icipa ed ia a ma hema ical pa h pu sued
in appendix D. Finally, in appendix E, we p esen esul s conce ning coope a ion le els and
AGoS a ained in mul ilaye s wi h di e en ypes o ne wo ks a e p esen ed.
7
2. LITERATURE REVIEW
In his sec ion we mo i a e he s udy o he p oblem o coope a ion. A li e a u e e iew is
unde aken om seminal pape s owa ds a gene al objec i e o assessing he cu en s a e o
knowledge on he subjec . The li e a u e e iew allowed us o iden i y di ec ions o esea ch
and unexplo ed ques ions. I also helped in adop ing a consis en e minology and on building
up he ounda ions o a heo e ical amewo k. Along his chap e , he li e a u e e iew is
p esen ed wi h an emphasis on a numbe o concep s ha a e he co ne s one o he esea ch
conduc ed and p esen ed in his hesis and ha se ed as he g ound/baseline o he ex ensi e
compu e simula ions conduc ed.
Impo an keywo ds ha summa ize he opics esea ched in his sec ion include
Coope a ion; Game Theo y; Complex Ne wo ks; E olu iona y Dynamics; Public Goods
Games; Mul ilaye Ne wo ks, and G adien o Selec ion.
2.1. THE PROBLEM OF COOPERATION
All g ea human achie emen s and he eme gence o human cul u e a e esul s o coope a i e
en e p ises. Genes coope a e o o m a genome, cells coope a e o p oduce mul icellula
o ganisms, indi iduals coope a e o o m g oups and socie ies. Language and human cul u e
a e jus examples o esul s om coope a i e en e p ises.
Coope a ion ela es o al uism, which opposes o compe i ion, a co ne s one o e olu ion
in Biology. Coope a ion can be iewed as an ou come o a game ha , despi e po en ial cos s
incu ed by pa icipa ing indi iduals, is “good” (measu ed by some app op ia ed i ness
measu e) o hem and ha equi es some so o collec i e ac ion. In his sense, o coope a e
means o beha e coope a i ely, o b ing some hing o he able.
The p oblem wi h coope a ion is ha equen ly i is cos ly, weighs on indi idual wellbeing
and p ospe i y and is, hus, always ulne able o exploi a ion by De ec o s. Indi iduals a e,
hus, di ided be ween ac ing sel ishly and sac i icing pa o hei sel in e es in exchange o
b inging alue o socie y.
The heo e ical amewo k used mos equen ly o s udy coope a ion among sel ish
indi iduals is E olu iona y Game Theo y (Nowak M. , 2006), whe e he concep o a social
dilemma cap u es he essence o he p oblem.
Social dilemmas o collec i e ac ion p oblems a e si ua ions whe e he e is a con lic
be ween indi idual and g oup in e es s so ha i he indi iduals y o maximize hei own
payo he whole g oup ends up wi h less han i hey had ac ed in ano he - ega ding way.
8
This ension be ween a ional choice and success ul coope a ion is he opic o a as li e a u e
sp ead o e disciplines o social science (E iksson & S imling, 2012).
Because coope a ion is bene icial o socie y, unde s anding he mechanisms and condi ions
ha encou age i and iden i ying he uning pa ame e s ha may in luence and ca alyse i s
eme gence is o u mos impo ance.
2.1.1. Two Pe son Games, he P isone ’s Dilemma
Classic game heo y is based on wo key assump ions. One is ha playe s ac a ionally,
hey a e ully awa e o hei and hei opponen ’s s a egy op ions and payo alues. They a e
capable o co ec ly assessing missing in o ma ion (i applicable) and p ocess new
in o ma ion e ealed by he play o opponen s (in dynamic games) in e ms o p obabili y
dis ibu ions.
The second assump ion is one o common knowledge, i.e., ha each playe knows no only
ha all o he s ac a ionally bu also ha o he s a e awa e ha he/she knows hey know, and
so on ecu si ely (Szabo & Fa h, 2007).
Playe s’ a ionali y assump ion has been elaxed o e ime in o de o push u he he
limi s o classic game heo y. Playe s ha e well de ined and consis en goals and p e e ences,
which can be desc ibed by a u ili y unc ion. A u ili y unc ion measu es he sa is ac ion
esul ing om a ce ain ou come o he game, and playe s’ goal is o maximize hei u ili y. I
mus be s essed he e ha he maximiza ion p oblem o game heo y di e s om a gene al
one o physics. In physics, one gene ally has a single pa ame ized unc ion whose ex eme
condi ion cha ac e izes he whole sys em. In game heo y, i is common o ha e ins ead a se
o unc ions o op imize as many as he numbe o in e ac ing playe s, as hey con inuously
es uc u e he landscape o each o he in pu sui o hei sel ish indi idual goals (op imum).
Playe s need o ha e a leas wo s a egies o choose om. The combina ion o s a egies
chosen by each playe , called s a egy p o iles, yields a esul . In classic game heo y, he
payo co esponds o he playe s’ e alua ions o his esul . A s a ic one-sho 2-playe game
can be ep esen ed as in able 3 by 𝑁 imes 𝑀 ma ices (no mal o m) wi h 𝜋1𝑖𝑗=𝑢1(𝑆1𝑖,𝑆2𝑗)
ep esen ing he u ili y unc ion o playe 1, when playe 1 and 2 apply 𝑆1𝑖 and 𝑆2𝑗 s a egies,
espec i ely. Recip ocally, 𝜋2𝑖𝑗
𝑇=𝑢2(𝑆1𝑖,𝑆2𝑗), whe e 𝑢2 ep esen s he u ili y unc ion o
playe 2 in he same ci cums ances.
Roles o he playe s de ine he game (a)symme y. In a symme ic game playe s’ oles a e
iden ical and in e changeable, which implies ha hey possess he same s a egy op ions and
payo s, N = M and 𝜋1𝑖𝑗=𝜋2𝑖𝑗.
9
Table 3 – Gene alized Payo Ma ix o a 2-Playe , 2-S a egy Game
Playe 2

(Payo )
𝑆21
…
𝑆2𝑀
Playe 1
𝑆11
(𝜋111,𝜋211
𝑇)
…
(𝜋11𝑀,𝜋21𝑀
𝑇)
⋮
⋮
⋱
⋮
𝑆1𝑁
(𝜋1𝑁1,𝜋2𝑁1
𝑇)
…
(𝜋1𝑁𝑀,𝜋2𝑁𝑀
𝑇)
A pa icula g oup o a symme ic wo-playe games, he only ones o be he subjec o his
hesis, happen when he numbe o pu e s a egies is 2, C (Coope a ion) and D (De ec ion),
wi h payo ma ix as in able 2.
In o de o be e cha and no malize e i o ies o ea and g eed in he playg ound o 2-
Pe son games and as p oposed in (San os, Pacheco, & Lenae s, 2006; San os F. C., Pinhei o,
Lenae s, & Pacheco, 2012) payo ma ix is linea ly ans o med in o de o and R and P o
alue 1 and 0, espec i ely.
Figu e 1- T-S Quad an s o 2-Playe Games
Ha ing applied his ans o ma ion and as depic ed in igu e 1, he S ag–Hun (SH) also
known as he Coo dina ion Dilemma eme ges wi h 𝑅>𝑇>𝑃>𝑆 and un olds in he lowe
le quad an , when he ea o being chea ed by 𝐷 (𝑃>𝑆) may jus i y de ec ion ins ead o
coope a ion.
In an isola ed en i onmen , each playe may decide o beha e as a C o a D. When bo h
playe s decide o coope a e, each ecei es a ewa d R. Mu ual de ec ion esul s in a
punishmen o P. I hey choose di e en s a egies, he one o e ing coope a ion ecei es S,
10
Sucke ’s payo , whe eas he De ec o collec s T, in e p e ed as he Temp a ion o de ec .
Di e en dilemmas esul om di e en o de ing o hese payo s.
Whene e 𝑇>𝑅, g eed eme ges and de ec ion is emp ing as i is he bes s a egy agains
a Coope a o . In he absence o ea (𝑃<𝑆), g eed leads o Chicken, Hawk-Do e o
Snowd i (SG) game wi h 𝑇>𝑅>𝑆>𝑃. Wi h bo h g eed and ea p esen , 𝑇>𝑅>𝑃>
𝑆, he game ob ained is P isone Dilemma (PD) (San os F. C., Pinhei o, Lenae s, & Pacheco,
2012).
Mo e gene ally, a game is a coope a i e dilemma when wo Coope a o s ge a highe
payo han wo De ec o s, 𝑅>𝑃, and s ill he e is an incen i e o de ec . This incen i e mus
exis s when a leas one o he ollowing condi ions hold ue: (i) i 𝑇>𝑅 hen i is be e o
de ec when playing agains a Coope a o ; (ii) i 𝑃>𝑆 hen i is be e o de ec when playing
agains a De ec o ; and (iii) i 𝑇>𝑆 hen i is be e o be he De ec o in an encoun e be ween
a Coope a o and a De ec o (Nowak M. , 2012).
This so o dilemma is p esen in anyone’s e e yday li e, when one has o decide be ween
commi ing o being lazy, being sel ish o al uis ic, e c.
A game is in a Nash equilib ium (NE) i , o each playe , he s a egy applied is he one
ha b ings him he highes payo conside ing he s a egies chosen by o he playe s, wi h is
he same o say ha he s a egy ollowed by each playe is he one he/she has no in e es o
de ia e om, as i maximizes his/he payo , aking in o accoun he s a egies chosen by
his/he pee s. The dilemma unde lying p isone ’s game is ha ac ing a ionally unable o
an icipa e opponen ’s s a egy, playe s will de ec and NE occu s wi h mu ual de ec ion,
al hough bo h playe s would be be e o i hey coope a ed.
NE is insensi i e o payo ma ix scaling o an addi ion o a bi a y cons an s o payo
columns (Szabo & Fa h, 2007). Thus, o se ing he ma ix in o de o ge a null payo when
De ec o s mee each o he and u he scaling i in o de o a 1 uni payo o esul be ween
Coope a o s in e ac ion as was pe o med in igu e 1, is a linea ans o ma ion om which
no loss o gene ali y in game dynamics esul s (B oom, 2005). The o se is i ele an in
eplica o dynamics based on payo di e en ial. As o he posi i e mul iplica i e ac o , i
only escales he ime.
Along he ex he concep o E olu iona y S able S a egy ela ed o he p obabili y o a
homogenous popula ion o be immune agains he in asion o a mino i y o in ude s o
mu an s will eme ge. A s a egy S is conside ed o be an E olu iona y S able S a egy (ESS)
i a popula ion composed only by S indi iduals is esilien o he in asion om a mino i y o
in ade s wi h any o he s a egy. Being a e inemen o a NE, no all NE a e necessa ily ESS.

11
2.1.2. Public Goods Games
The e a e many socially and economically impo an examples wi h a numbe o decision
make s in ol ed g ea e han wo. Al hough some imes hese si ua ions can be modelled as
epea ed play o simple pai in e ac ions, he e a e many cases whe e he mos undamen al
uni o he game is i educibly o mul i-playe na u e. These games canno be cas in a ma ix
o bi-ma ix o m. S ill he basic solu ion concep is he same: when played by a ional agen s
he ou come should be a Nash equilib ium whe e no playe has an incen i e o de ia e
unila e ally.
One example o Public Goods Games (PGG) is he T agedy o he Commons, an abs ac
game ha exempli ies wha has been one o he majo conce ns o poli ical philosophy and
economic hinking since he 19 h cen u y (Ha din, 1968) and is now pa o he mains eam
economic heo y, assuming ha he sel ish and a ional human na u e will lead o he deple ion
o essen ial and common esou ces, e.g. wa e , soil, e c., in he absence o well-de ined
p ope y igh s, o mal, op-down managemen ins i u ions, ules o access and exploi a ion.
In o de o quan i a i ely be e illus a e how he T agedy o Commons un olds, one can
assume a common ini e esou ce, e.g. a illage g een, and N playe s, a me s. The cos o
aking one goa g azing in he g een is c. I is up o each playe i o decide on how many goa s
𝑔𝑖 he/she will ake g azing. In o al he e will be 𝐺=𝑔1+⋯+𝑔𝑁 goa s g azing. u(G) is an
u ili y unc ion ha e u ns he indi idual bene i om aking a goa g azing as a unc ion o
g een u iliza ion. As he g een su e s om o e g azing, no only will 𝑢(𝐺) dec ease wi h G,
which means 𝑑𝑢
𝑑𝐺<0, as his dec ease will be sha pe o highe G, which means 𝑑2𝑢
𝑑𝐺2<0. The
payo o each playe will be o
𝑝𝑖=(𝑢(𝐺)−𝑐)𝑔𝑖
(1)
A a Nash equilib ium, ame s’ decision on he numbe o goa s g azing will be (𝑔1∗,…,𝑔𝑁
∗)
as no a me will be be e o i he changes his/he chosen numbe o goa s in he g een. Thus,
a Nash equilib ium 𝜕𝑝𝑖
𝜕𝑔𝑖=0, i.e.,
𝜕𝑝𝑖
𝜕𝑔𝑖=𝑢(𝐺∗)+𝑑𝑢(𝐺∗)
𝑑𝐺 𝜕𝐺
𝜕𝑔𝑖𝑔𝑖∗−𝑐=𝑢(𝐺∗)+𝑑𝑢(𝐺∗)
𝑑𝐺 𝑔𝑖∗−𝑐= 0
(2)
Summing up le iden i y o igh mos equa ion o all playe s, one ge s
𝑢(𝐺∗)+𝐺∗
𝑁𝑑𝑢(𝐺∗)
𝑑𝐺 −𝑐=0
(3)
12
I 𝑢(𝐺) is known, he op imum 𝐺∗ can be ound. Howe e , we e he e a cen al managemen
en i y in place, he social wel a e would ake place a 𝐺∗∗which maximizes o al payo 𝑝=
(𝑢(𝐺)−𝑐)𝐺. This implies
𝑢(𝐺∗∗)+ 𝐺∗∗𝑑𝑢(𝐺∗∗)
𝑑𝐺 −𝑐= 0
(4)
Compa ing bo h equa ions and aking in o accoun he ac o 𝑢(𝐺) and i s de i a e dec easing
wi h 𝐺, one concludes ha 𝐺∗∗<𝐺∗, which means ha a Nash equilib ium compa ed wi h
social wel a e op imum he common esou ce is o e u ilized. This makes he game a social
dilemma (Szabo & Fa h, 2007).
In ano he a ia ion o PGG (Kleinebe g & Helbing, 2018; Ba is on, Ma jaz, & La o a,
2017; Li, Shen, & Jiang, 2016; Pacheco, Pinhei o, & San os, 2009; San os, San os, & Pacheco,
2008) each playe 𝑖 in a o al o 𝑁 makes a con ibu ion 𝑐𝑖 o a common pool, opped a bi a ily
by 1. The o al collec ed, ∑𝑐𝑖𝑖 is mul iplied by an enhancemen ac o 𝐹,1<𝐹<𝑁, a
syne gy ac o e lec ing how much he whole is g ea e han he sum o he pa cels, o be
equally di ided among all pa icipa ing playe s, no ma e he amoun o indi idual
con ibu ions,. Being a De ec o in his game means con ibu ing wi h 𝑐𝑖=0. Those who
con ibu e wi h 𝑐𝑖>0 a e Coope a o s.
Maximum o al income is achie ed i all playe s con ibu e maximally. In his case each
playe ecei es 𝐹𝑐, esul ing in a inal payo is (𝐹−1)𝑐. Playe s a e aced wi h he
emp a ion o being ee- ide s, i.e., o ake ad an age o he common pool wi hou
con ibu ing o i , as any indi idual in es men is a loss o he playe because only he ac ion
𝐹
𝑁<1 will be epaid. Consequen ly, a ional playe s in es no hing and one ends up wi h
ano he T agedy o Commons, F ee Ride p oblem, Social Dilemma on N-Playe PD (Szabo
& Fa h, 2007).
I he numbe o playe s is 2 and playe s’ choices a e bina y, i.e., hey a e cons ained o
no in es o o in es a ixed amoun , hen he game becomes a P isone ’s Dilemma. Wi h
due uning o pa ame e s, a N-pe son ound obin PD game can simula e a PGG.
As i will be he ea e discussed in Ne wo k sec ion indi iduals a e loca ed as nodes in a
ne wo k. Each PGG ins ance makes use o a ocal node such ha he ocal node has all i s
di ec neighbou s, i.e., all nodes di ec ly linked o he ocal node, cons i u e he N playe s o
he game.
While i is common o assume ha in e e y game indi iduals can con ibu e/in es a ixed
amoun c o he public good, a b oade scena io inspi ed in he 2-Playe Dis ibu ed P isone
Dilemma (DPD) om (Pacheco, Pinhei o, & San os, 2009) is explo ed. Hence, a PGG
13
in ol ing wo indi iduals ha pa icipa e in mul iple games is conside ed, in es men alues
being dis inc pe playe s. In he dis ibu ed scena io, indi iduals ha e o spli hei in es men
ac oss a se o games hey pa icipa e ( ha can be all o hem, o pa o hem). In ha case,
he possible ou comes o each playe ac ions can be summa ized in a payo ma ix as in able
4 wi h 𝐶1 and 𝐶2 ep esen ing, espec i ely, he in es men s o playe s 1 and 2.
Table 4 - Payo ma ix o Public Goods Games in he 2-Playe Dis ibu ed
(Ve sion in Playe 1’s Pe spec i e)
𝜋
(Payo )
Playe 2
C
D
Playe 1
C
(𝐹2−1)𝐶1 + 𝐹2 𝐶2
(𝐹2−1)𝐶1
D
𝐹2𝐶2
0
Depending on he assump ions o he DPD he alues o 𝐶1 and 𝐶2 may be compu ed
di e en ly. The single N+1-Playe game is subs i u ed by N 2-Pe son games, one pe each o
he N neighbou s he ocal node can play wi h.
2.1.3. E olu iona y dynamics in ini e popula ions
The e olu iona y game dynamics o a ini e popula ion can be desc ibed by a s ochas ic
p ocess, an app oach well sui ed o compu e simula ion ha models he mic oscopic
mechanisms unde lying s a egy ans e ence be ween indi iduals. Once he e olu iona y pa h
is aced, one will o ce bo h (i) popula ion size o end o in ini y and (ii) ime in e als
be ween sys em upda es o end o ze o, looking o a con e gence wi h he solu ion ha would
ha e been eached had he popula ion been conside ed in ini e and dynamical ules de ined a
popula ion le el.
As a mechanism o s a egy ans e ence, some al e na i es can be conside ed om which
he ollowing a e highligh ed:
 Pai wise compa ison- Along his al e na i e, a ocal indi idual a ailable o upda e
his s a egy is andomly selec ed. A second dis inc indi idual is also andomly
selec ed. All indi iduals ha e an equal p obabili y o being chosen in any selec ion.
14
The i s indi idual copies a s a egy o a second indi idual wi h a p obabili y ha
inc eases wi h he i ness di e en ial be ween hem. The e e ence p obabili y can be
P ob =12+𝑤2𝜋𝑟−𝜋𝑓
△𝜋
(5)
wi h △𝜋 ep esen ing he maximum payo di e ence ha can be ound be ween
indi iduals, he nume a o o he ac ion s anding o he di e ence be ween and
indi iduals’ payo and 0≤𝑤≤1 . 𝑤 measu es he ela i e impo ance o selec ion
compa ed o neu al d i .
An al e na i e o no ha ing o an icipa e △𝜋 is o ely on Fe mi dis ibu ion and
ha e (T aulsen, Nowak, & Pacheco, 2006)
P ob =1
1+𝑒−𝑤(𝑛𝑟−𝑛𝑓)
(6)
In bo h pai wise compa isons, o 𝑤=0 he decision o upda e a s a egy has 0.5
p obabili y and does no ake in o accoun payo di e ences. Fo 0<𝑤≪1, he wo
e e ence p obabili ies become simila because 1
1+𝑒−𝑤(𝑛𝑟−𝑛𝑓) =12+ 𝑤
2(𝑛𝑟−𝑛𝑓)+
𝑂(𝑤2). Speci ically o he Fe mi case, i 𝑤→∞ he p ocess becomes de e minis ic:
an indi idual swi ches s a egy i and whene e he one he compa es o has an highe
payo (T aulsen & Haue , 2008).
Pai wise compa ison models a p ocess o cul u al e olu ion by lea ning and imi a ion.
 Mo an bi h-dea h p ocess- Fi s ly a ocal indi idual is andomly selec ed o
ep oduc ion wi h a p obabili y p opo ional o i s i ness. His/He o sp ing inhe i s
ances o ’s s a egy. Ano he dis inc indi idual is selec ed andomly wi h uni o m
p obabili y. In o de o p ese e he size o he popula ion, he indi idual is eplaced
by he o sp ing. The Mo an bi h-dea h p ocess (Mo an, 1958) o igina ed in gene ics
p o ides a mechanism o mos i indi iduals o sp ead ac oss he popula ion. Fi ness
o an indi idual as p oxyed by i s payo can be gi en by 1−𝑤+𝑤𝜋, wi h 0≤𝑤≤
1 s anding o he balance be ween selec ion and neu al d i .
The Mo an bi h-dea h p ocess maps o he adi ional in e p e a ion o e olu iona y
game dynamics in which s a egies a e encoded in genomes and sp ead h oughou he
popula ion as a unc ion o i s ela i e i ness.
In o de o be e illus a e how ini eness impac s e olu ion, one s a s conside ing a well-
mixed popula ion wi h N indi iduals and wi h wo s a egies, A and B, a ailable o
indi iduals o choose om wi h a gene ic payo ma ix 𝜋𝑖𝑗 as in able 5.
21
𝜕
𝜕𝑡𝜌(𝑥,𝑡0)=−𝜕
𝜕𝑥[(𝑇+(𝑥0)−𝑇−(𝑥0))𝜌(𝑥0,𝑡0)]+
12𝜕2
𝜕𝑥2[(𝑇+(𝑥0)+𝑇−(𝑥))𝜌(𝑥0,𝑡0)]
𝑁 + 𝑂(𝑁−2)
(36)
𝑥0 and 𝑡0 a e poin s wi hin in e als ]𝑥,𝑥+1
𝑁[ and ]𝑡,𝑡+1
𝑁[, espec i ely, wi h 𝑁 ending o
in ini y. In o de o sol e his equa ion, a pa en hesis is opened o de i e he same equa ion
ia a S ochas ic Di e en ial equa ion o he o m
𝑑𝑋𝑡=𝜇(𝑥,𝑡)𝑑𝑡+𝜎(𝑥,𝑡)𝑑𝐵𝑡
(37)
𝑋𝑡 is a s ochas ic p ocess wi h a d i o e ime o 𝜇(𝑥,𝑡) and a local ola ili y gi en by 𝜎(𝑥,𝑡).
Bo h 𝜇 and 𝜎 unc ions a e de e minis ic. 𝐵𝑡 is a B ownian p ocess, also known as Wiene
p ocess, esul ing om he in eg a ion o whi e noise. I is cha ac e ized by being s a iona y
wi h 𝑃𝑟𝑜𝑏(𝐵0=0)=1 and 𝐵𝑡−𝐵𝑠~𝑁(0,|𝑡−𝑠|). Di e en ial calculus will be o no use
he e as 𝐵𝑡, al hough con inuous, is no di e en iable. Howe e , I ô lemma can be applied.
Fu he calcula ing densi y p obabili y 𝑝(𝑥,𝑡) o p ocess X, one ge s a simila Fokke -Planck
equa ion (Oksendal, 2003)
𝜕
𝜕𝑡𝑝(𝑥,𝑡)=−𝜕
𝜕𝑥(𝜇(𝑥,𝑡)𝑝(𝑥,𝑡))+12𝜕2
𝜕𝑥2(𝜎2(𝑥,𝑡)𝑝(𝑥,𝑡))
(38)
Compa ing equa ion 36 o 38 and conside ing p ocess X as de ined in 37 o be a solu ion o
equa ion 38, a solu ion o equa ion 36 is
𝑑𝑥(𝑡)
𝑑𝑡 =(𝑇+(𝑥)−𝑇−(𝑥))+√𝑇+(𝑥)+𝑇−(𝑥)
𝑁𝜂(𝑡)
(39)
Second e m on he igh hand side, which includes a whi e noise componen 𝜂(𝑡) de i a e o
he B ownian p ocess can be disca ded, because o 𝜂(𝑡) ha ing a no malized Gaussian
ampli ude p obabili y and he popula ion size 𝑁 ending o in ini y. Payo o ollowe s o A
and B s a egy om a S ochas ic Ma ko Bi h-Dea h p ocess as in equa ions 7 and 9 adap ed
o he con inuous case and s ill conside ing payo as a p oxy o i ness esul s in
𝑓𝐴(𝑥)=𝑎𝑥+(1−𝑥)𝑏
𝑓𝐵(𝑥)=𝑐𝑥+(1−𝑥)𝑑
(40)
(41)
𝑇±(𝑥) o he Fe mi dis ibu ion pai wise compa ison, he one o be explo ed along he hesis,
alues
𝑇±(𝑥)= 𝑥(1−𝑥) 1
1+𝑒±𝛽(𝑓𝐵(𝑥)−𝑓𝐴(𝑥))
(42)
Fo 𝑇+(𝑥), 1−𝑥 s ands o he p obabili y o i s indi idual o be andomly selec ed om B
popula ion, 𝑥 o he p obabili y o second selec ed indi idual o be om popula ion A and

22
inally he ac ion ep esen s he Fe mi p obabili y o i s indi idual o copy s a egy om
second one.
In oducing equali ies om equa ion 40 o 42 in o equa ion 39 and conside ing in his las
equa ion only he i s e m om igh side, as second one is disca dable when N ends o
in ini y, one ge s
𝑑𝑥(𝑡)
𝑑𝑡 = 𝑥(1−𝑥)(1
1+𝑒𝛽(𝑓𝐵(𝑥)−𝑓𝐴(𝑥))−1
1+𝑒−𝛽(𝑓𝐵(𝑥)−𝑓𝐴(𝑥)))=
𝑥(1−𝑥) anh (𝛽2(𝑓𝐴(𝑥)−𝑓𝐵(𝑥)))
(43)
which o o 𝛽≪1 simpli ies o
𝑑𝑥(𝑡)
𝑑𝑡 =𝛽2𝑥(1−𝑥)(𝑓𝐴(𝑥)−𝑓𝐵(𝑥))=𝛽2𝑥(𝑓𝐴(𝑥)−〈𝑓(𝑥)〉)
(44)
wi h 〈𝑓(𝑥)〉 s anding o he a e age payo o he popula ion, equal o 𝑥𝑓𝐴(𝑥)+(1−
𝑥)𝑓𝐵(𝑥).
This is he eplica o equa ion, a de e minis ic equa ion o in ini e popula ions o be
in oduced in he ollowing chap e s a ing ha he ela i e g ow h a e o a popula ion
(1𝑥𝑑𝑥(𝑡)
𝑑𝑥 ) is p opo ional o he di e en ial o i s i ness o a e age i ness. I assumes
indi iduals a e equally likely o in e ac wi h any o he s. (T aulsen, Claussen, & Haue , 2006;
T aulsen & Haue , 2008) show ha he eplica o equa ion is also he limi o equa ion 36 o
in ini e popula ions o Mo an bi h-dea h p ocess.
2.1.4. The Replica o Dynamics
On a la ge uns uc u ed popula ion ending o in ini y, he ules desc ibing he selec ion
among a limi ed numbe o s a egies is de ined a mac oscopic le el. They assume he o m
o nonlinea di e en ial equa ions coined as he eplica o dynamics ha ake in o
conside a ion he selec ion mechanism applicable, modelling he e olu ion o he popula ions’
equency by means o i ness compa isons. The eplica o equa ion p o ides a mean- ield
de e minis ic desc ip ion o a popula ion e olu iona y dynamics, which means conside ing an
in ini e and well-mixed popula ion d i en by a con inuous ime dynamical p ocess.
Conc e izing, by aking a dynamic pe spec i e in E olu iona y Game Theo y and by
in e p e ing he a e o ep oduc ion o a popula ion as i s i ness, a popula ion wi h x( )
indi iduals a ime wi hou en i onmen cons ain s agains i s g ow h and ep oducing a a
a e o pe indi idual and uni o ime has an e olu ion o e ime ha can be desc ibed by
he di e en ial equa ion
23
𝑑𝑥
𝑑𝑡 = x
(45)
wi h he solu ion
x( ) = 𝑥0𝑒𝑟𝑡
(46)
whe e 𝑥0 ep esen s he size o he popula ion a 𝑡=0. I 𝑟 is posi i e, he popula ion g ow hs
o in ini y. I popula ion dea h is o be conside ed, he di e en ial equa ion is o be eplaced
by
𝑑𝑥
𝑑𝑡=(𝑟−𝑑)𝑥
(47)
wi h 𝑑 ep esen ing he dea h a e pe indi idual and uni o ime.
I one conside s addi ionally a maximum en i onmen ca ying capaci y 𝐾, g ow h a e can
be opped by a (1−𝑥
𝐾) ac o as in
𝑑𝑥
𝑑𝑡 = (𝑟−𝑑)𝑥(1−𝑥
𝐾)
(48)
wi h solu ion
𝑥(𝑡)= 𝐾𝑥0𝑒(𝑟−𝑑)𝑡
𝐾+𝑥0(𝑒(𝑟−𝑑)𝑡−1)
(49)
So a , a single popula ion o indi iduals was conside ed. When second popula ion is
in oduced, na u al selec ion ge s in he play because i is a key mechanism o e olu ion ha
ope a es whene e di e en ypes o indi iduals ep oduce a di e en a es.
Wi h mo e han one popula ion and conside ing a scena io in which he o al popula ion is
held cons an , e.g. due o he ecosys em ha ing a maximum cons an ca ying capaci y, on he
mac oscopic le el he eplica o dynamics can be pos ula ed di ec ly wi h he easonable
assump ion ha he pe capi a g ow h a e o a popula ion, 1𝑥𝑑𝑥
𝑑𝑡, is p opo ional o he
popula ion i ness o a e age i ness di e en ial (Szabo & Fa h, 2007; Nowak M. , 2006),.
The i ness measu es he indi idual’s e olu iona y success, i.e., he payo o he game in his
game heo y con ex .
Along his a ional, le 𝑥 and 𝑦 ep esen he a io o indi iduals o e o al popula ion and
le 𝑎 and 𝑏 s and o ep oduc ion a ios o X and Y popula ions, espec i ely. Ob iously
𝑥 + 𝑦 = 1. Le also ∅=ax+by s and o he a e age i ness o he all popula ion. Then,
acco ding o assump ions abou popula ions’ g ow h, one has
𝑑𝑥
𝑑𝑡 = 𝑥(𝑎−∅)
(50)
(51)
24
𝑑𝑦
𝑑𝑡 = 𝑦(𝑏−∅)
Because he sum o 𝑥 and 𝑦 p opo ions is ixed, he sum o hei de i a e equals 0. So, i igh
sides o bo h equa ions a e added one ge s (ax + by) − (x+y)∅, which equals ze o as
expec ed. P e ious sys em o equa ions is edundan , so eplacing y by 1 –𝑥 in i s equa ion
(o ice- e sa) one ge s
𝑑𝑥
𝑑𝑡 = 𝑥(1−𝑥)(𝑎−𝑏)
(52)
Equilib ium is eached when 𝑑𝑥
𝑑𝑡=0, i.e., when 𝑥=0 o 𝑥=1. This makes sense, because i
co esponds o all popula ion consis ing only o X o Y indi iduals. The equa ion highligh s
ano he aspec . I 𝑎>𝑏, 𝑑𝑥
𝑑𝑡>0, which implies ha X popula ion will domina e and Y be
ex inc . I 𝑎<𝑏 i is Y ime o domina e. I 𝑎=𝑏, wha e e he ini ial ela i e p opo ions
o X and Y popula ion, hey a e p ese ed.
The e olu iona y scena io analysed is an example o he su i al o he i es , bu o he
scena ios can be an icipa ed in which bo h popula ion can co-exis . Such scena ios can be
ep esen ed by mo e gene al e olu iona y equa ions such as
𝑑𝑥
𝑑𝑡 = 𝑎𝑥𝑐−∅𝑥
𝑑𝑦
𝑑𝑡 = 𝑏𝑥𝑐−∅𝑦
(53)
(54)
whe e 𝑥, 𝑦, 𝑎, and 𝑏 main ain hei p e ious meaning. The e is now a new a iable, 𝑐. I 𝑐=
1, he p e ious scena io is eco e ed. In o de o keep o al popula ion cons an , i.e., o he
sum o a ia ions in X and Y popula ions o be ze o, ∅ is upda ed o ∅=𝑎𝑥𝑐+
𝑏𝑦𝑐. Subs i u ing ∅ in i s sys em o equa ions one ge s
𝑑𝑥
𝑑𝑡 = 𝑎𝑥𝑐−(𝑎𝑥𝑐+𝑏𝑦𝑐)𝑥 =
𝑥(𝑎𝑥𝑐−1−𝑎𝑥𝑐−𝑏𝑦𝑐)=
𝑥((1−𝑥)𝑎𝑥𝑐−1−𝑏(1−𝑥)𝑐−1 =
𝑥(1−𝑥)(𝑎𝑥𝑐−1−𝑏(1−𝑥)𝑐−1)
(55)
The de i a e o X popula ion equency is ze o o 𝑥=0 o 𝑥=1, as in p e ious scena io.
The e is howe e a new oo o his de i a e ha alues
𝑥∗= 1
1+ √𝑎𝑏
𝑐−1
(56)
Depending on he combina ion o 𝑎, 𝑏 and 𝑐 pa ame e s, his oo can be s able o uns able.
25
Figu e 3- Examples o supe exponen ial and subexponen ial g ow hs. G ow h is go e ned by 𝑑𝑥
𝑑𝑡 =
𝑥(1− 𝑥)(𝑎𝑥𝑐−1−𝑏(1−𝑥)𝑐−1) equa ion. Le panel ob ained wi h 𝑎=9, 𝑏=2, 𝑐=3 has an uns able oo .
In igh panel, he combina ion 𝑎=3, 𝑏=8, 𝑐=0.5 leads o a s able oo .
Figu e 3 p esen s examples o bo h so s o oo s. The oo 𝑥∗ na u ally belongs o in e al
]0,1[. Howe e , depending on 𝑐− 1 signal, he beha iou o he sys em changes. I 𝑐 > 1,
he de i a e o X popula ion equency will be nega i e in in e al ]0,𝑥∗[, posi i e in ]𝑥∗,1[.
I he de i a e is posi i e, he end o he popula ion is o g ow. Thus, in i s in e al he
popula ion sh inks, o g ow on he second one.
No ma e how small he pe u ba ion is, i 𝑥 su passes 𝑥∗, because o 𝑑𝑥
𝑑𝑡 being posi i e, 𝑥
keeps inc easing un il 𝑥=1. The ecip ocal happens i a pe u ba ion makes 𝑥 smalle han
𝑥∗, wi h 𝑥 da ing o 0. This makes 𝑥∗ is an uns able equilib ium poin .
The ema kable aspec o s ess is ha his conclusion is i espec i e o he a es o g ow h
o X and Y popula ions. E en i Y has a highe ep oduc ion a e, i a a ce ain poin in ime
equency o popula ion X goes beyond 𝑥∗, popula ion Y is doomed.
I 𝑐<1, 𝑥=0 and 𝑥 = 1 a e s ill equilib ium poin s bu uns able. The in oduc ion o a
minimum numbe o indi iduals om X o Y popula ion in a popula ion o all Y o all X
indi iduals, espec i ely, d i es he sys em o a co-exis ence scena io wi h 𝑥∗ and 𝑦∗ =
1 – x∗ as he ela i e equencies o X and Y popula ions, espec i ely. Again, i espec i e
o he ep oduc ion a es o bo h popula ions. The ela i e ep oduc ion a es o he
popula ions only de e mines whe e x* is loca ed.
Because s a ing ela i e equency dic a es he s able equilib ium poin o con e ge o,
supe exponen ial g ow h (𝑐>1) a ou s whoe e was he e i s (su i al o he i s ),
whe eas subexponen ial g ow h (𝑐 < 1) leads o he su i al o all. This is illus a ed in
igu e 3.
26
2.2. THE MECHANISMS OF COOPERATION
Coope a ion p oblems eme ge as a esul om a misalignmen be ween indi idual
mo i a ions and collec i e goals. This ension is bes cap u ed by he p isone 's dilemma a
2-Playe game wi h 𝑇>𝑅 >𝑃 >𝑆, played in bo h e i o ies o g eed, as i is emp ing o
de ec agains a Coope a o because 𝑇 > 𝑅, and ea as wi h 𝑃 > 𝑆 de ec ion a ises as he
bes s a egy agains a De ec o (see igu e 1). Thus, no ma e he s a egy ollowed by he
opponen , he bes op ion o a a ional playe is always o de ec . By seeking sel ishly o
maximize his/he own p o i s, a ional playe s end up in a less desi able collec i e ou come,
ins ead o 2𝑅 hey ge 2𝑃.
Na u al selec ion is a key mechanism o e olu ion ha ope a es whene e di e en ypes
o indi iduals ep oduce a di e en a es (Nowak M. , 2006) which leads one o expec ha
e e y indi idual should be designed o p omo e i s own e olu iona y success a he expense
o i s compe i o s. This is why, in he absence o any o he assump ion and in a well‐mixed
popula ion, De ec o s always ha e a highe expec ed payo han Coope a o s, and he e o e
na u al selec ion ewa ds sel ish beha iou and a ou s De ec o s.
In opposi ion, coope a ion is an al uis ic ac ha is cos ly o pe o m, because i means
indi iduals gi ing up pa o hei ( ep oduc i e) po en ial in o de o bene i s o he s and in
a ou o a common good
Besides, coope a ion is always ulne able o exploi a ion by De ec o s. Thus, his s a ed,
coope a ion u u e does no look p omising.
Ye coope a ion is obse ed on many le els o biological o ganiza ion, om bac e ia and
cellula o ganisms o animals (Nowak M. , 2007). Coope a ion is he decisi e o ganizing
p inciple o human socie y. Many g ea achie emen s o humankind we e accomplished ia
coope a ion. Addi ionally, popula ions o De ec o s ha e a lowe i ness han i hey played
he coope a ion ole.
So, besides elying on indi idual social alue o ien a ion (Bogae , Boone, & Decle ck, 2007),
a concep om social psychology a emp ing o e lec how much weigh a pe son a aches o
he wel a e o o he s in ela ion o i s own, o jus i ying coope a ion, explaining he

27
Figu e 4- Mechanisms o Coope a ion. Clockwise om op le co ne di ec ecip oci y mechanism is
p esen ed wi h indi iduals epea edly in e ac ing and helping each o he . Indi ec ecip oci y ollows. By
helping a pee an indi idual builds on his epu a ion. In kin selec ion he le el o help depends on pee
ela edness. In g oup selec ion when a an indi idual ep oduces in a g oup al eady a i s maximum capaci y he
g oup spli s in wo and ano he al eady exis ing g oup is chosen o ex inc ion. Popula ion s uc u e dic a es
who in e ac s wi h whom allowing he o ma ion o clus e
eme gence o coope a ion equi es ce ain mechanisms in place o na u al selec ion o a ou
coope a ion o e de ec ion and o p e en Coope a o s om losing g ound o De ec o s. Such
mechanisms will ha e o su e o o se he cos s o coope a ion by causing Coope a o s o
also be on he ecei ing end mo e o en.
In (Nowak M. , 2007) 5 such mechanisms a e iden i ied: kin selec ion, di ec ecip oci y,
indi ec ecip oci y, g oup selec ion and ne wo k ecip oci y, which a e schema ically
ep esen ed in igu e 4. Fo each mechanism, he au ho s s a om a PD payo ma ix duly
adap ed wi h new pa ame e s in o de o i o co ec ly e lec he in e ac ion be ween wo
basic s a egies wi h he mechanism in ac ion. In doing so ewa ds om choosing a
coope a ion s a egy ge mo e appealing. Condi ions a e e en c ea ed o coope a ion o
become an e olu iona y s able s a egy. A six h mechanism o punishmen was also
conside ed as s udied in (Feh & Gäch e , 2002; Fowle & Ha pending, 2005).
These mechanisms can be b ie ly summa ized as ollows:
 Kin Selec ion- Kin selec ion ope a es whene e in e ac ions occu among gene ic
ela i es, i.e., among indi iduals who a e mo e p obable o sha e a common ances o
28
han i hey we e andomly sampled om he whole popula ion. Rela edness o
indi iduals is de ined as he p obabili y o hem sha ing a gene. The coe icien o
ela edness be ween wo indi iduals, 𝑟, a alue in he in e al [0,1] equals 1/2 o wo
b o he s, 1/8 o cousins (Nowak M. , 2007).
 Di ec Recip oci y- In na u e coope a ion be ween un ela ed indi iduals is no iceable,
so kin selec ion ails sho o explain coope a ion in mo e gene al ci cums ances. In
o de o ackle his limi a ion, (T i e s, 1971) p oposed di ec ecip oci y as ano he
mechanism o he e olu ion o coope a ion based upon he p inciple o “I will help
you i you help me la e ”. The ac o encoun e s be ween he same playe s been
epea ed u ns coope a ion mo e in i ing.
 Indi ec Recip oci y- Di ec ecip oci y elies on epea ed encoun e s be ween he
same wo indi iduals. Help p o ided by he dono is less cos ly han bene icial o he
ecipien . Howe e , pa icula ly in human ela ions, in e ac ions a e asymme ic and
unbalanced. Indi ec ecip oci y elies on epu a ion and applies mos ly o human
ela ionships. “Fo di ec ecip oci y one needs a ace, bu o indi ec ecip oci y a
name is needed ins ead”. Indi ec ecip oci y is build ou o di ec ecip oci y
wi nessed by an in e es ed audience. Encoun e s a e obse ed by o he s and
in o ma ion sp eads h ough communica ion channels, allowing indi iduals o adop
condi ional s a egies depending on he epu a ion o he pee in he game. Di ec
ecip oci y elies on a playe ’s own expe ience wi h someone, while indi ec
ecip oci y uses he expe ience o o he playe s. In indi ec ecip oci y he help
p o ided may ne e be e u ned by he bene icia y, o by indi iduals who in u n ha e
been helped by he bene icia y (Nowak M. S., 1998).
 G oup Selec ion- G oup selec ion also known as mul ile el selec ion is based on he
assump ion ha compe i ion occu s no only be ween indi iduals bu also be ween
g oups. (T aulsen & Nowak, 2006) p opose a minimalis s ochas ic model o g oup
selec ion whe e he popula ion is subdi ided in o 𝑛𝑔 g oups, which g ow in size as
indi iduals wi hin hem ep oduce. In any one ime s ep, a single indi idual om he
en i e popula ion is chosen o (gene ic) ep oduc ion wi h a p obabili y p opo ional
o i s payo . The o sp ing is added o he same g oup. When a g oup eaches a
h eshold size N, i ei he di ides in o wo child g oups wi h p obabili y q (in which
case a andom g oup om he popula ion is elimina ed), o i does no di ide (wi h
complemen a y p obabili y 1−𝑞), in which case a andom indi idual in he g oup is
29
elimina ed so ha g oups do no ge o e popula ed. Social in e ac ions occu only
among membe s o he same g oup and indi iduals bea ing a mu an allele, a pa icula
o m o a gene, help o he s by dec easing hei payo by 𝑐 as he coun e pa o
gene a ing a bene i 𝑏 o be sha ed by all o he g oup membe s. As a esul , sel ish
indi iduals end o eplica e as e han helpe s wi hin g oups, bu g oups comp ising
helpe s g ow as e and ha e a g ea e chance o di iding be o e isking ex inc ion.
 Ne wo k Recip oci y - Wi h no mechanisms o ca alyse coope a ion, na u al selec ion
a ou s de ec ion because a well-mixed popula ion is assumed whe e a playe
po en ially can play any o he wi h equal p obabili y. This app oxima ion is used by
all s anda d app oaches o e olu iona y game dynamics (Nowak M. , 2007). Howe e ,
in eali y popula ions a e no well-mixed and i is no equally likely ha a playe mee s
any o he . In ac indi iduals end o in e ac wi h a limi ed numbe o pee pe
geog aphic easons o any o he s. Thus, wi h ne wo k ecip oci y popula ions a e
suppo ed in ne wo ks wi h indi iduals a hei nodes, links ep esen in e ac ions and
de e mine who can in e ac wi h whom. In his con ex , i has been shown ha
coope a ion may eme ge (o no ) depending on he opology o he in e ac ion g aph
(San os & Pacheco, 2005; San os, Pacheco, & Lenae s, 2006). As discussed in mo e
de ail below, he ne wo k s uc u e changes he e ec i e game played a a popula ion-
wide le el, e en i , locally, indi iduals con inue o ace he same dilemma (Pinhei o,
Pacheco, & San os, 2012). The ne wo k ecip oci y mechanism elies on wo ac o s.
The i s is a limi a ion in he numbe o game opponen s, ha is, “dep essing
anonymi y,” a he han ha ing an in ini e and well-mixed popula ion (Oh suki,
Haue , Liebe man, & Nowak, 2006). Second one is a local adap a ion mechanism, in
which a playe can only copy a s a egy om a di ec ly linked neighbou as de e mined
by unde lying ne wo k (Tanimo o, 2015).
 Punishmen - F om an e olu iona y pe spec i e, coope a ion is a double-edged swo d.
On he one hand, i b ings an edge ad an age o a communi y, since some asks can
only be achie ed h ough coope a ion. On he o he hand, since punishmen in ol es
addi ional cos s (Feh & Gäch e , 2002; Fowle & Ha pending, 2005) om an
indi idual's pe spec i e i becomes emp ing o enjoy he esul s o coope a ion,
wi hou in es ing in i . This is he ypical ee- ide p oblem ha cha ac e izes social
dilemmas and i allowed o p oli e a e can b eak down coope a ion. Since selec ion in
e olu ion akes place on he le el o he indi idual, Coope a o s a e eplaced by ee-
ide s, pu ing coope a ion o an end. In o de o discou age ee- ide s beha iou , a
30
mechanism o punishmen is conside ed wi h wo main objec i es: i s one is o expel
om he g oup ee- ide membe s, om which decision a payo pe capi a inc ease
esul s; second one is o accoun o he cos s o a g oup exclusion in de ec ion s a egy,
which ends up as a dissuasi e measu e.
A mo e exhaus i e lis o suppo ing mechanisms o he e olu ion o coope a ion can be
ound in (Zaggi, 2013).
O all hese coope a ion mechanisms conside ed, his hesis ocus on ne wo k ecip oci y.
2.3. THE SCIENCE OF NETWORKS
A ne wo k in i s simples o m is a collec ion o poin s joined oge he in pai s by lines.
Poin s a e e e ed o as nodes o e ices (V) and ep esen he elemen s in a sys em, e.g.
s a ions in subway map, and he lines, e e ed o as links o edges, ep esen a di ec ela ion
be ween he nodes hey connec , e.g. a line connec ion be ween wo s a ions.
Many objec s o in e es in he physical, biological, and social sciences can be hough o
as ne wo ks. F om a modelling s andpoin , a ne wo k is a ela i ely simple objec , consis ing
o only nodes and links o mally desc ibed by 𝐺 = (𝑉,𝐸) whe e 𝑉 ep esen s he se o nodes
and 𝐸

𝑉 𝑥 𝑉 he se o links conside ed as pai s o nodes linked oge he .
Social ne wo ks is a ecen ja gon e e ing o he mesh o social ela ionships be ween
indi iduals in a g oup, communi y o popula ion. The ype o ela ionships dic a es he ype
o ne wo k, be i scien i ic collabo a ion, p o essional, hobby-o ien ed, e c. The ubiqui y o
ne wo ks a ex end social domain, o en e hose o anspo a ion /subway, ai lines), powe
g id, elecommunica ions, heal h ca e and o he s (Kim, Ola e-Rojas, Ál a ez-Mi anda, &
Seung-Woo, 2018).
In gene al, no only a e ne wo ks shaped om indi iduals’ ac ions as ecip ocally
indi iduals’ ai s and beha iou a e la gely in luenced by he ne wo k (Gi a d, He , &
Schunk, 2014). In he pa icula case o social ne wo ks, his indi idual e sus collec i e
beha iou had al eady been add essed decades ago in Sociology wi h S uc u a ion heo ies
which eme ged as an a emp o dispel di ision wi hin he social sciences be ween hose who
conside ed social phenomena o be de e mined by objec i e social s uc u es (de e minism)
and o he s who saw social phenomena as he ou come o human agen s subjec i ely
in e p e ing he wo ld ( olun a ism) (Timb ell, Delaney, Chan, Yue, & Gable, 2005).
37
N (see 2.3.1.1), a he han ollowing a polynomial expec ed o egula la ices.
 High Clus e ing
The a e age clus e ing coe icien o eal ne wo ks is much highe han he one
expec ed o a andom ne wo k o simila numbe o nodes and links.
Figu e 8- F om Regula o Random Newo ks. (sou ce: (Wa s & S oga z, 1998)). L is de ined as he
numbe o edges in he sho es pa h be ween wo e ices, a e aged o e all pai s o nodes. C measu es
clus e ing coe icien a e aged o e all nodes. A p obabili y o 10−3is enough o making a e age dis ance in
he ne wo k o decay a ound 45%. A p obabili y o 10−2al eady makes a e age dis ance o decay 80% wi h
almos no deg ada ion o clus e ing index.
As in igu e 8, one s a s wi h a egula ing la ice whe e nodes a e posi ioned. Each node
is connec ed o <𝑘>
2 neighbou s on each side, wi h 〈𝑘〉 deno ing he in ended a e age deg ee.
I 〈𝑘〉 is odd i is a bi a ed which side has egula ly minus one neighbou s.
In a second s ep, o each node 𝑛1, he connec ion o each o i s 𝑛2 igh side neighbou s is
conside ed o ewi ing wi h p obabili y p, a pa ame e o he algo i hm. In case o ewi ing,
a node 𝑛3 is andomly chosen wi h uni o m p obabili y among hose wi h which 𝑛1 has no
ye a connec ion and excluding 𝑛1. Rewi ing consis s o 𝑛1 connec ing o 𝑛3 and
disconnec ing om 𝑛2.
The Wa s-S oga z ne wo k in e pola es be ween a egula la ice, which has high
clus e ing bu lacks he small-wo ld phenomenon, and a andom ne wo k, which has low
clus e ing, bu displays he small-wo ld p ope y.
Because o he small-wo ld cha ac e inhe i ed om andom ne wo k in luence ia he
ewi ing mechanism ac i a ed, Wa s-S oga z ne wo ks a e also known as small-wo ld
ne wo ks .
This small-wo ld e ec had al eady been iden i ied by S anley Milg am in an expe imen
conduc ed in he la e six ies ha was la e coined as six deg ees o sepa a ion. Su p ising
Regula
Random
Small-Wo ld
(Wa s-

38
esul s showed ha despi e socie y huge size o 6 billion indi iduals, by ollowing social links
any pai o nodes is on a e age six links apa , equi ing each pe son o ha e an a e age deg ee
lesse han 2 (Ba abási & F angos, Linked, 2002).
2.3.1.2. Ho and Ne wo ks
In a homogeneous ne wo ks such as la ices, all nodes sha e he same deg ee implying
deg ee dis ibu ion o be a del a unc ion exhibi ing a single peak a 〈𝑘〉.
This use ul ea u e is combined wi h he small-wo ld one in Ho and ne wo ks (San os,
Rod igues, & Pacheco, 2005; San os F. C., Pinhei o, Lenae s, & Pacheco, 2012). The
cons uc ion o a Ho and ne wo k as in Wa s-S oga z model s a s wi h a egula ing la ice
whe e nodes a e posi ioned. Being 〈𝑘〉 he in ended deg ee o he ne wo k, each node connec s
o he closes <𝑘>
2 nodes on i s le and igh side. Up o now, he ne wo k exhibi s he in ended
deg ee, has maximum clus e ing coe icien bu lacks small-wo ld ea u e. Second s ep o
Wa s-S oga z algo i hm p o ides he small-wo ld cha ac e bu a expenses o deg ee
a ia ion among nodes. To acqui e he small-wo ld ea u e wi hou sac i icing deg ee
homogenei y a pai o links is chosen andomly. Subjec o a ce ain p obabili y p, one node
o one link is exchanged wi h one node in he o he link, ensu ing no duplica ion o links
among he same pai o nodes. The p ocess is epea ed i e a i ely un il a ac ion p o all links
has been ewi ed.
The appea ance o a Ho and ne wo k is simila o a Wa s-S oga z one bu wi h a ixed
deg ee dis ibu ion.
2.3.1.3. Scale-F ee Ne wo ks
The deg ee dis ibu ion o a andom ne wo k is o Poisson o m, which means ha mos
nodes ha e a deg ee ha is close o i s a e age. I implies addi ionally ha he e a e ex emely
ew nodes highly connec ed, because he cu e alls away om i s peak as e han
exponen ially.
The andom model o E dős-Rényi es s on wo simple and o en dis ega ded assump ions.
The i s one is ha he se o nodes i ixed, emains unchanged h oughou he ne wo k
li e ime and is known up on om he beginning o ne wo k concep ion. The second one is
ha all nodes a e equal. Unable o dis inguish be ween he nodes, hey link andomly o each
o he (Ba abási & F angos, 2002).
39
Figu e 9- Poisson e sus Powe -law Dis ibu ions. (Sou ce: (Ba abási & Pós ai, Ne wo k Science, 2016))
On he le panel o igu e 9 a Poisson dis ibu ion o he p obabili y o inding a node wi h
deg ee k (𝑝𝑘) om andom ne wo ks is compa ed o a powe -law dis ibu ion (γ= 2.1) om
Scale- ee ne wo ks on a log-log plo . Bo h dis ibu ions ha e ⟨k⟩= 11.
On he cen al and igh panel a andom and a scale- ee ne wo ks a e plo . Bo h ne wo ks
ha e 50 nodes and 〈𝑘〉=3. The size o each node is p opo ional o i s deg ee.
(Reka, Jeong, & Ba abasi, 1999) s udied he Wo ld Wide Web and ound ou ha deg ee
dis ibu ion ollows a powe law, whe e he p obabili y o a node ha ing deg ee 𝑘 is
p opo ional o 𝑘−

wi h


2. The same conclusion was d awn o o he ne wo ks analysed
as powe -g id, Hollywood ne wo k o ac o s o IBM chip wi ing diag am. A Scale-F ee (SF)
ne wo k is a ne wo k whose deg ee dis ibu ion ollows a powe law as in igu e 9. The SF
quali ie de i es om he ac al-like cha ac e o he ne wo k, namely ha he dis ibu ion o
he nodes’ deg ee is p ese ed no ma e he scale o analysis. Thus, o popula ions ending
o in ini y, he dis ibu ion o he node deg ee in in e al [a, b], apa om a mul iplica i e
ac o , is in a ian whene e in e al bounda ies a e mul iplied by ano he posi i e ac o .
The second and highe momen s o he deg ee dis ibu ion goes o in ini y when  < 3. Fo
many SF ne wo ks,  is loca ed be ween 2 and 3.
This ype o dependency on k, which is no exclusi e o he echnology ield o online
communi ies bu also cha ac e izes o he ne wo ks in he na u al wo ld and social ields, is
no o ui ous bu de i es ins ead om he p ocess conduc ing o ne wo k g ow h o e ime.
In (Ba abási, 2013) ou o ganizing p inciples a e iden i ied o ne wo ks such as web and
social ne wo ks o be held oge he . SF is he i s o ganizing p inciple, which implies ha
hubs a e no only ole a ed, bu expec ed. Small-wo ld is he second p inciple, which s a es
ha wo nodes a e likely o be connec ed by a ela i ely sho pa h o nodes, e en in a e y
Scale-F ee
Random
40
la ge and spa se SF ne wo k as he Web. The hi d p inciple is o p e e en ial a achmen by
which newcome s o he ne wo k p e e ably connec o nodes wi h highe deg ee. The o h
p inciple is ela ed o he no ion o i ness and allows me i o p e ailed o e senio i y and
p o ide compe i ion. Ne wo k g ow h de elops o e ime, new links a e s ablished be ween
old nodes, bu he a e o g ow h is con olled by he i ness and he nodes wi h a g ea e i ness
will end o ‘win ou ’ and become e y highly connec ed. This explains why Google o
Facebook a i ed la e o he web bu became winne s. By iewing ne wo ks as dynamical
sys ems ha change con inuously o e ime, he SF model embodies a new modelling
philosophy.
SF opology is he esul o o ganizing p inciples ac ing a each s age o he ne wo k
o ma ion p ocess. G ow h and p e e en ial a achmen explain he basic ea u es o many o
he ne wo ks seen in na u e, social ne wo ks included. As long as hese ing edien s a e p esen ,
i will main ain i s hub-domina ed SF opology.
One impo an p ope y o SF ne wo ks is esilience agains andom e o . I nodes a e
emo ed a andom om mos ypes o ne wo ks hey will e en ually agmen in o a se o
smalle ne wo ks o indi idual nodes, bu ins ead a SF ne wo k will emains obus agains
andom decay; i may sh ink bu no all apa . I e en emains connec ed inde ini ely i  < 3
as in he case o Wo ld Wide Web. The Achilles’ heel o a SF ne wo k is an in en ional a ack
a ge ing he mos connec ed nodes (Ba abási, 2013).
(Ba abási, 1999) p opose an algo i hm o gene a e a SF ne wo k wi h  = 3 used ex ensi ely
along he simula ions in his hesis. I s a s wi h 𝑚0 nodes ully connec ed. Then i e a i ely
nodes a e added. Each new node is connec ed o 𝑚 dis inc nodes (𝑚

𝑚0) al eady pa o
he ne wo k. A new node choses andomly a node in he ne wo k o connec o wi h a
p obabili y p opo ional o i s deg ee. This mechanism a ou s senio i y and c ea es hubs. In
o de o a oid a co ela ion be ween he index o he nodes and i s deg ee, a inal s ep once
he ne wo k is cons uc ed consis s o sc ambling he indexes o he nodes. The a e age deg ee
o c ea ed ne wo k ends o 〈𝑘〉=2𝑚. SF ne wo ks wi h

=3 a e called Ba abási-Albe .
Ba abási-Albe s esses he small-wo ld cha ac e is ic o he ne wo k by ha ing
〈𝑑〉~ln (𝑁)
𝑙𝑛(ln(𝑁))
(73)
2.3.1.4. Mul ilaye s
Ne wo ks consis ing o a se o nodes o e ices connec ed by links o edges ha e been
ypically used o desc ibe o example a ic in a ci y, in e ac ions be ween indi iduals, he
41
ade among ma ke s o he Wo ld Wide Web. In o ma ion in he o m o news, messages, o
digi al i uses can be ansmi ed h ough ne wo ks, as well as in ec ious diseases o gene al
goods. Se e al algo i hms ha e been p oposed o gene a e and eplica e he mos impo an
s uc u al p ope ies o eal-wo ld ne wo ks om which examples we e p esen ed in p e ious
chap e s.
In spi e o he p og ess achie ed in ne wo k science du ing ecen decades, adi ionally
one has assumed ha nodes a e connec ed o each o he wi hin he same, isola ed
in as uc u e, he so-called single-laye ne wo k.
This assump ion, howe e , may in some ci cums ances be an o e simpli ica ion,
conside ing ha some nodes can simul aneously be he building blocks o mo e han jus one
ne wo k. This impo an conside a ion applies o na u al as well as o social sys ems. Many
complex sys ems demand mani old esou ces o be supplied om dis inc channels o unc ion
p ope ly, such as wa e , gas, and elec ici y o a ci y (De Domenico, e al., 2013).
As majo ci ies a e in e connec ed no jus by means o oads, bu also by means o ails,
as well as by means o ai anspo a ion, simila ly, people in e ac ace- o- ace, ia phone, on
online social ne wo ks, in hei wo k en i onmen , and so on (Min, 2014). I is hus o en
jus i ied o abandon he adi ional assump ion o a single-laye ne wo k and eplace i wi h a
mul ilaye ne wo k o malism. No su p isingly hen, he mul ilaye ne wo k, de ined as a
combina ion class o ne wo ks ha a e in e ela ed in a non i ial way, has ecen ly eme ged
as a undamen al concep o quan i a i ely desc ibe he in e ac ions no jus wi hin, bu also
among di e en ne wo ks.
Ne wo ks o ne wo ks ha e been b ough o he spo ligh by he disco e y ha e en small
and seemingly i ele an changes in one ne wo k can ha e ca as ophic and e y much
unexpec ed consequence in ano he ne wo k (Buldy e , Pa shani, Paul, S anley, & Ha lin,
2010).
Wha se s a mul ilaye ne wo k apa om he adi ional single-laye ne wo k is ha a
mul ilaye ne wo k ypically consis s o 𝑀 (𝑀 > 1) ne wo ks (o laye s), whe e he nodes in
each ne wo k (laye ) a e connec ed ia in a-laye links, bu he e may be also in e -laye links
ha connec oge he nodes om o he ne wo ks. Some imes he in e -laye links do no se e
o connec he nodes, bu me ely se e o communica e in o ma ion o some o he o m o
in luence be ween he nodes o ming he 𝑀 ne wo ks.
42
The same node may appea in mo e han one ne wo k and some imes all he nodes pe ain
o all 𝑀 ne wo ks wi h he di e ence be ween hem being he in a-laye links. Depending on
hese pa icula i ies, he e minology ha is used also a ies.
One o he challenges in ne wo k heo y is he e o e o ea oge he ies o di e en kind
p ese ing exis ing di e ences. The mul ilaye me apho , which allows o dis inguish he
di e en kinds o ela ionships among a se o nodes, cons i u es a p omising amewo k o
s udy and model mul ilaye sys ems.
I e olu iona y games a e played on mul iplex ne wo ks, s a egy imi a ion and payo
accumula ion can ake place ei he in he local neighbou hood o a pa icula laye o ac oss
he laye s, since all nodes exis in all laye .
2.3.1.5. De ini ion o a Mul ilaye Ne wo k
When nodes a e connec ed o each o he in a single in as uc u e one has a single-laye
ne wo k o g aph o mally desc ibed by G = (V, E) whe e V ep esen s he se o nodes and E
 V x V he se o links conside ed as pai s o nodes linked oge he .
Howe e , a single-laye can be an o e simpli ica ion, e.g. i nodes ep esen ai po s and
he links he di e en ai line companies ligh s. Ai lines ha e hei own se o connec ions
linking he ai po s, cons i u ing a laye o co e age independen om he ones co esponding
o o he companies. Whene e he e a e di e en in as uc u es o conside and he e a e cos s
lowing be ween hem, i is p e e able o conside a mul ilaye o mally desc ibed as 𝐺𝑀=
(𝑉𝑀,𝐸𝑀) whe e 𝑀 ep esen s he numbe o laye s, 𝑉𝑀= 𝑈𝛼=1
𝑀 𝑉𝛼 𝑤𝑖𝑡ℎ 𝑉𝛼={𝑉1𝛼,…,𝑉𝑁𝛼
𝛼}
and 𝑁𝛼 equal o he numbe o nodes in laye α. 𝑉𝛼 and 𝑉𝑀 ep esen , espec i ely, he se o
nodes in laye α and in he all mul ilaye .
In wha conce ns mul ilaye edges one has 𝐸𝑀 = {𝐸𝛼 𝑈 𝐸𝛼

:𝛼,

𝜖 {1,…,𝑀},𝛼 ≠

}
wi h 𝐸𝛼⊆ 𝑉𝛼𝑥 𝑉𝛼 and 𝐸𝛼𝛽⊆ 𝑉𝛼𝑥 𝑉𝛽.
Acco ding o (Wang, Wang, Szolnoki, & Pe c, 2015) and as in igu e 10 mul ilaye s can
be classi ied in o h ee b oad ca ego ies:
 Mul iplex Ne wo ks
In a mul iplex ne wo k all he laye s con ain he same se o nodes o sha e a leas
some ac ion o he nodes. The di e ence be ween he laye s is he way he nodes a e
connec ed wi h each o he in each pa icula laye .
The ne wo k o ai po s can be ansla ed in o a mul iplex o m wi h di e en laye s
consis ing o he ou es o di e en ai plane ca ie s. The collabo a ion and he ci a ion
ne wo ks cons i u e ano he example o laye s o a mul iplex.

43
Figu e 10- Mul ilaye Ne wo ks. Mul iplex ne wo k ha e he same nodes ac oss all laye s. In e dependen
ne wo ks ypically ha e di e en nodes in di e en laye s. Addi ionally well-being o nodes in one laye may
depend o he well-being o o he nodes in o he laye s. In e connec ed ne wo k a e like in e dependen ones
bu wi h physical connec ions in e -laye . In all he h ee ypes o mul ilaye s in a-laye connec ions a e
independen ac oss laye s.
All laye s ha ing he same nodes and no in e -laye links can be o mally exp essed as
𝑉𝛼 ∩ 𝑉

= 𝑉𝑀 = 𝑉,∀ 𝛼 ≠

. Addi ionally, Eα =∅,∀ α ≠  , i.e., no links
be ween laye s.
 In e dependen Ne wo ks
An in e dependen ne wo k is ypical made up o wo o mo e di e en ne wo ks, such
ha he e is li le o no o e lap be ween he nodes in he di e en laye s. The
pe o mance o nodes in a pa icula laye depend on he pe o mance o nodes in a
di e en laye , and ice e sa. Thus, he e exis s he so-called dependency links
be ween he nodes ha a e pa o di e en laye s. These links a e no ac ual physical
links, bu a he imagina y links ha deno e he co-dependence; hence he name
in e dependen ne wo ks.
The concep o in e dependen ne wo ks is e e ed o in (Buldy e , Pa shani, Paul,
S anley, & Ha lin, 2010) whe e cascading ailu es be ween an elec ical g id ne wo k
and a compu e ne wo k a e s udied. Ai po s and seapo ne wo ks can be in e p e ed
as in e dependen ne wo ks, because he p ope unc ioning o a seapo may depend
on goods deli e ed by ai .
44
Each laye ha ing di e en ypes o nodes and he e being no dependency links (no
physical connec ions) be ween he nodes can be o malized as 𝑉𝛼 ∩ 𝑉

= ∅ ∀ 𝛼≠
𝛽.
 In e connec ed ne wo ks
An in e connec ed ne wo k is simila o an in e dependen ne wo k in ha i is ypical
made up o wo o mo e di e en ne wo ks, such ha he e is li le o no o e lap
be ween he nodes in he di e en laye s. In he in e connec ed ne wo k, howe e ,
he e a e ac ual physical links ha connec oge he he nodes om di e en laye s.
In e connec ed ne wo ks can hus be ega ded as in e connec ed communi ies o
clus e s wi hin a single la ge ne wo k.
The clima e ne wo k can be decomposed in o di e en ne wo k in e connec ed laye s
in exploi ing he s a i ica ion and ci cula ion o he e es ial a mosphe e.
Each laye ha ing di e en ypes o nodes and he e being ac ual physical links
be ween he nodes in di e en laye s can be o malized as ∃𝛼,𝛽∈ {1,…,𝑀} 𝑉𝛼 ∩
𝑉

≠ ∅ ⋀ 𝛼≠ 𝛽.
Real mul iplex ne wo ks a e a om andom supe posi ion o hei cons i uen laye
opologies. Ins ead, he deg ees o he same nodes in di e en laye s may be co ela ed, as
hey may end o connec o simila nodes in di e en laye s c ea ing o e lapping edges.
In his hesis only mul iplex ne wo ks will be add essed. Whene e he ea e mul ilaye
social ne wo k a e e e ed o, i is i s mul iplex a ian o be assumed.
2.3.1.6. Deg ee-Deg ee Co ela ion
An in e es ing p ope y obse ed in eal mul iplex ne wo ks is he p esence o co ela ions
be ween he deg ees o he same node a di e en laye s. This is no mally signalled by he
ac ha he p obabili y 𝑃(𝑘𝛼= 𝑘11,𝑘

= 𝑘2) o ind a node wi h deg ee 𝑘1 on laye α and
deg ee 𝑘2 on laye  does no ac o ize in he p oduc 𝑃𝛼(𝑘)𝑃

(𝑘) o he deg ee dis ibu ions
o he wo laye s.
Wi hin a laye , deg ee co ela ion cap u es he ela ionship be ween he deg ees o nodes
linking o each o he . Laye s can be asso a i e, i nodes wi h a highe (lowe ) deg ee end o
link o nodes wi h a highe (lowe ) deg ee, disasso a i e i he end is he o he way a ound
o neu al i no end is iden i ied.
Mul iplex ne wo ks, because o ha ing he same nodes in all laye s, allow he ex ension o
in a-laye deg ee co ela ion concep o mul iplex scope. This is he a ionale behind deg ee-
45
deg ee co ela ion index designed o quan i y asso a i e (disasso a i e) mixing pa e n
be ween laye s. The deg ee-deg ee co ela ion is e iden in social ne wo ks. I a amous
indi idual like a singe o a spo sman is amous in one ne wo k, e.g. Facebook o Twi e ,
p obabili y he/she will also be amous in ano he one. Deg ee-deg ee co ela ion is supposed
o e lec his asso a i e pa e n.
(Nicosia & La o a, 2015) p opose se e al me hods o calcula e deg ee-deg ee co ela ions.
One possibili y is he Pea son's linea co ela ion coe icien . I 𝑘𝑖𝛼and 𝑘𝑖𝛽 deno e espec i ely
de deg ee o node 𝑖 in laye 𝛼 and 𝛽, Pea son's co ela ion coe icien o he wo-deg ee
sequences is de ined as:
𝑟𝛼𝛽 = 1
𝑁∑𝑘𝑖𝛼𝑘𝑖𝛽
𝑁
𝑖=1 − 1
𝑁2∑𝑘𝑖𝛼∑𝑘𝑖𝛽
𝑁
𝑖=1
𝑁
𝑖=1
√1
𝑁∑(𝑘𝑖𝛼)2−(1
𝑁∑𝑘𝑖𝛼
𝑁
𝑖=1 )2
𝑁
𝑖=1 √1
𝑁∑(𝑘𝑖𝛽)2−(1
𝑁∑𝑘𝑖𝛽
𝑁
𝑖=1 )2
𝑁
𝑖=1
(74)
which can succinc ly be s a ed as
𝑟𝛼𝛽 = <𝑘𝑖𝛼𝑘𝑖𝛽>− <𝑘𝑖𝛼><𝑘𝑖𝛽>
𝜎𝑘𝛼𝜎𝑘𝛽
(75)
Ano he possibili y is o use he Spea man's ank co ela ion coe icien , which o he wo
𝑅𝑖𝛼 and 𝑅𝑖𝛽 ank sequences o deg ee in laye α and β, espec i ely, is gi en by:
𝜌𝛼,𝛽=∑(𝑅𝑖𝛼−𝑅𝛼




)(𝑅𝑖𝛽−𝑅𝛽




)
𝑖
√∑(𝑅𝑖𝛼−𝑅𝛼




)2∑(𝑅𝑗𝛽−𝑅𝛽




)2
𝑗𝑖
(76)
whe e 𝑅𝑖𝛼 and 𝑅𝑖𝛽a e he anks o node i due o i s deg ee in laye s α and β, espec i ely, and
𝑅𝛼




and 𝑅𝛽




a e he a e age anks o nodes in he same laye s
1
.
A hi d al e na i e would be o use Kendall’s 𝜏 ank co ela ion coe icien .
(Nicosia & La o a, 2015) calcula e he 3 co ela ion coe icien s o Ame ican Physical
Socie y co-au ho ship ne wo k wi h 10 laye s each one assigned o sub ields o esea ch a ea.
The p ocedu e is epea ed o IMDb ne wo k o collabo a ion be ween ac o s wi h laye s
de ined acco ding o mo ie gen e. In bo h ne wo ks, he 3 co ela ion coe icien s p o ide a
consis en indica ion abou asso a i eness be ween laye s.
(Nicosia & La o a, 2015) p o ide wo algo i hms based on simula ed annealing o cons uc
mul iplex ne wo ks wi h con ollable in e -laye deg ee-deg ee co ela ion, use ul o
1
Wi hin laye 𝛼, node 𝑖 ha ing ank 𝑅𝑖𝛼 means ha in his laye nodes wi h an highe
deg ee han 𝑖’s ha e 𝑅𝑖𝛼−1 dis inc alues o deg ee. E.g., nodes wi h second g ea e deg ee
in a laye ha e ank 2 in ha laye .
46
simula ions in o de o be e unde s and how deg ee co ela ions ac oss laye s in luence
mul iplex pe o mance.
2.3.1.7. O e lapping
In mul iplexes, ne wo ks nodes ake pa in all laye s o ne wo ks simul aneously. Social
ne wo ks whe e each indi idual node has di e en kind o social ies, one o amily ones,
ano he o iendship, o p o essional, e c., o anspo a ion sys ems whe e each loca ion is
connec ed o ano he loca ion by di e en ypes o anspo , one pe laye , a e jus examples
o mul iplex cha ac e ized by a signi ican o e lap o he links in di e en laye s (Bianconi,
2013).
O e lapping be ween wo laye s 𝛼 and 𝛽 p o ides an indica o on how p obable i is o an
a bi a y pai o nodes o be linked in bo h laye s. I is de ined acco ding o (Ba is on, Ma jaz,
& La o a, 2017) as
o e lapping𝛼𝛽= ∑𝑎𝑖𝑗
𝛼𝑎𝑖𝑗
𝛽
𝑖<𝑗
∑𝑎𝑖𝑗
𝛼+ ∑𝑎𝑖𝑗
𝛽
𝑖<𝑗 − ∑𝑎𝑖𝑗
𝛼𝑎𝑖𝑗
𝛽
𝑖<𝑗𝑖<𝑗
(77)
wi h 𝑎𝑖𝑗
𝛼 and 𝑎𝑖𝑗
𝛽 ep esen ing, espec i ely, he adjacency ma ixes in laye s 𝛼 and 𝛽 and
𝑎𝑖𝑗
𝛼= {1, i nodes 𝑖 and 𝑗 a e linked in laye 𝛼
0, o he wise
(78)
O e lap be ween wo laye s alls in he in e al [0,1]. Minimum alue is achie ed i laye s
ha e no links in common. Maximum alue e lec s he case o bo h laye s coinciding. Deg ee-
deg ee co ela ion is a mo e elaxed measu e o he simila i y be ween laye s, because a
co ela ion o 1 does no imply laye s’ coincidence. I only conce ns he same node on
di e en laye s ha ing he same numbe o neighbou s no ma e i hey di e be ween laye s.
The o e lapping o he whole mul iplex can be de ined as he a e age o he o e lap o
e e y pai o laye s:
o e lapping = 1
𝐶2𝑀∑o e lapping𝛼𝛽
𝑀
𝛼=1,𝛽=1,𝛼<𝛽 =
2
𝑀(𝑀−1)∑o e lapping𝛼𝛽
𝑀
𝛼=1,𝛽=1,𝛼<𝛽
(79)
2.4. EVOLUTIONARY GAMES ON STRUCTURED POPULATIONS
E olu iona y Game Theo y is a ma hema ical app oach aiming o desc ibe he compe i ion
o species in an ecosys em, he in e ac ion be ween indi iduals o dis inc popula ions,
whe he hey a e molecules, li ing o ganisms o humans ideas and beha iou s. The
mechanism ha leads o he e olu ion o coope a ion in hese se ings could be called ‘spa ial
53
Nowak, 2006; T aulsen, Sho esh, & Nowak, 2008), ailing o explain how a s a egy co-exis s
o ou compe es ano he . The weakness o analy ical s udies s em om hei alidi y scope,
o en equi ing un ealis ic popula ion s uc u es o o he ex eme condi ions such as weak-
selec ion o Mo an p ocesses, which means ha ing 𝑤≪1 when i ness (𝑓) is ela ed o
payo (𝜋) as in
𝑓=(1−𝑤)+𝑤𝜋
(83)
Besides, hese s udies a ge on ixa ion p obabili ies, a concep speci ic o biology, missing
he nexus be ween he ne wo k opology and he e olu iona y dynamics c ea ed. Un il
(Pinhei o, Pacheco, & San os, 2012), ne wo k s udies ailed o cha ac e ize he sel -o ganizing
p ocess by which one s a egy co-exis s wi h o displaces he o he .
In o de o shed some ligh on he opological mechanism ha depa ing om ne wo ked
indi iduals engaged in a local game leads o a global, popula ion wide, beha iou al dynamics
which de ia es s ongly om he o iginal one, (Pinhei o, Pacheco, & San os, 2012) de ine an
a e age g adien o selec ion (AGoS) o ack he sel -o ganiza ion o Coope a o s when
in e ac ing wi h De ec o s unde ne wo k ecip oci y. A any poin in ime, AGoS e lec s he
expec ed inc ease in he numbe o Coope a o s in he popula ion. In a bi a y ne wo k
popula ions, i is no possible o de i e a closed o m exp ession o he AGoS, bu i can be
nume ically calcula ed. I allows assessing he ole o popula ion s uc u e in he e olu iona y
dynamics since i is by design simila o he eplica o equa ion in an in ini e well-mixed
popula ion.
Fo AGoS calcula ion, e olu ion is modelled ia a s ochas ic bi h-dea h p ocess whe e
each indi idual 𝑖 wi h payo 𝑝𝑖 adop s he s a egy o a andomly selec ed neighbou 𝑗 wi h
payo 𝑝𝑗 wi h p obabili y gi en by he Fe mi unc ion
𝑝= 1
1+𝑒−𝛽(𝑝𝑗−𝑝𝑖)
(84)
Fo each indi idual i, he p obabili y o imi a ing he beha iou o any o i s neighbou s a
ime 𝑡, 𝑇𝑖(𝑡), is gi en by
𝑇𝑖(𝑡)= 1
𝑘𝑖∑1
1+𝑒−𝛽(𝑝𝑗−𝑝𝑖)
𝑛𝑖
𝑗=1
(85)
wi h 𝑘𝑖 and 𝑛𝑖 ep esen ing, espec i ely, he deg ee o 𝑖 and he numbe o i s neighbou s
wi h a di e en s a egy hen he one om 𝑖.

54
I 𝑇𝑖(𝑡) is a e aged ac oss all Coope a o s o De ec o s, one ob ains he p obabili y o he
popula ion a ime 𝑡 o dec emen o inc emen by 1 he numbe o Coope a o s, espec i ely,
𝑇𝐴−(𝑐,𝑡) and 𝑇𝐴+(𝑐,𝑡), gi en by
𝑇𝐴±(𝑐,𝑡)= 1
𝑁∑ 𝑇𝑗(𝑡)
𝑁
𝑗=1,𝐴𝑙𝑙𝐷𝑠
𝑗=1,𝐴𝑙𝑙𝐶𝑠
(86)
The i s pa ame e in 𝑇𝐴±(𝑐,𝑡) was added o make explici he dependency o his a iable
on he numbe o Coope a o s in he popula ion.
Du ing he cou se o simula ion s a ime , he expec ed inc ease in he numbe o
Coope a o s in he popula ion al eady wi h c Coope a o s is gi en by
𝐺𝑠(𝑐,𝑡)=𝑇𝐴+(𝑐,𝑡) - 𝑇𝐴−(𝑐,𝑡)
(87)
a quan i y wi h an absolu e alue no g ea e han 1.
T us ing on he e godici y o he p ocess, which means ha a e ages o e p obabili y space
and o e ime coincide, AGoS is ob ained by a e aging 𝐺𝑠(𝑐,𝑡) ac oss he ime o a la ge
numbe o samples:
AGoS(c) = 1
ΛΩ∑ ∑ 𝐺𝑠(𝑐,𝑡)
∈ Ωs ϵ Λ
(88)
whe e Λ ep esen s he space dimension o all ne wo k e olu iona y simula ions. Ω ep esen s
he numbe o ime s eps pe simula ion.
In a ne wo k wi h N nodes, he e a e 𝐶𝑐𝑁= 𝑁!
(𝑁−𝑐)!𝑐! dis inc con igu a ions wi h c
Coope a o s. Along he e olu ion simula ions, some s a egy con igu a ions will be mo e
likely han o he s, e.g. mo emen s om unco ela ed asso men s o co ela ed ones should
be mo e p obable han he o he way ound, which will make he p obabili ies o he a ious
con igu a ions wi h c Coope a o s unequal.
In o de o AGoS o be d awn o all possible c numbe o Coope a o s and o a maximum
numbe o dis inc con igu a ions wi h c Coope a o s o be e alua ed, each simula ion in he
Λ space is o be ini ialized wi h Coope a o s andomly posi ioned in he ne wo k in a numbe
gi en by a uni o mly dis ibu ed a iable in he in e al [0, N], in o de o p e en lack o
obse a ions in he s a e space.
AGoS being a opology dependen mean- ield desc ip o o a s uc u ed popula ion can be
compu ed o a bi a y popula ion s uc u e and a bi a y game pa ame e iza ion.
Figu e 13 exempli ies possible shapes o AGoS.
55
Figu e 13-A e age G adien o Selec ion
A e age G adien o Selec ion in s uc u ed popula ions a e measu ed ac oss ne wo ks
andomly sampled om a se o ne wo ks o he same ype, e.g. Ho and s BA, deg ee,
asso a i i y, e c.
Clockwise, s a ing in op le co ne in i s panel, he ne wo k will e ol e owa ds Full
Coope a ion. In nex panel he e olu ion is owa d Full De ec ion. In nex panel, he ne wo k
is bound o ei he Full Coope a ion o Full De ec ion, inal des ina ion depending on ini ial
Coope a o equency in ela ion o c i ical equency x*. One has a Coo dina ion Game. In
las panel, he sys em con e ges o a s a egy co-exis ence scena io. The ype o ne wo k o
he ela i e alues o payo ma ix con ibu e o dic a e he ype o AGoS applicable.
2.4.4. The S uc u e o Social G aphs
The connec i i y s uc u e can be cha ac e ized by a numbe o opological p ope ies
p esen ed in sec ion 2.3. The g aphs o conside along he hesis a e all connec ed, meaning
he e is a leas one pa h along links be ween any pai o nodes.
Ne wo ks co-exis in di e en laye s o a mul iplex. All laye s sha e he same se o nodes,
bu wi h a laye speci ic link se . Nodes in di e en laye s may ha e deg ee-deg ee co ela ion
o links ha e a ce ain amoun o o e lapping be ween laye s.
Squa e la ices a e one o he simples ne wo ks. They can be a la ice wi h on Neumann
neighbou hood, 4 connec ions be ween nea es neighbou nodes, o Moo e neighbou hood
wi h connec ions be ween nea es and nex -nea es neighbou s, in a o al o 8, e c. I may be
he case ha egula la ice only p o ides an ini ial s uc u e o he c ea ion o mo e ealis ic
56
social ne wo ks whe e a po ion o playe s and/o in e ac ions a e andomly emo ed (Nowak
M. , 2006).
La ices lack he small-wo ld ypical cha ac e is ic o eal-ne wo ks. Along he hesis,
Ho and ne wo ks will be conside ed. Due o hei cons uc ion p ocess no only do hey exhibi
he small-wo ld cha ac e is ics bu also a ixed deg ee, use ul when ying o unde s and
analy ically ce ain phenomena o he ole o ano he p ope y, wi hou deg ee a ia ion.
Homogenous ne wo ks, in which ca ego y Ho and ne wo ks i in, allow he c ea ion o
clus e s o Coope a o s ha in e ac and maximize payo in o de o be e esis De ec o s
exploi a ion, al hough wi h an high sensi i i y on he in ensi y o selec ion (Pinhei o, Pacheco,
& San os, 2012).
A pa e n obse ed in nume ical simula ions is ha o games played on ne wo ks, a he
beginning he numbe o Coope a o s dec eases, as a De ec o playing agains a Coope a o
ge s a highe payo . Howe e , on he long un, small clus e s o Coope a o s o m on he
la ice, because Coope a o s playing agains each o he pe o m be e han De ec o s agains
De ec o s. The ini ial dec ease in coope a ion is he p ice o pay o Coope a o s o eo ganize
and assemble (Pe c, 2013).
Fo Mo an p ocess o dea h-bi h ype in a dona ion game wi h weak selec ion (see equa ion
83), (Oh suki, Haue , Liebe man, & Nowak, 2006) s a e ha coope a ion is a ou ed
whene e 𝑏𝑐>〈𝑘〉, wi h b s anding o he bene i o an al uis ic ac , c o i s cos and 〈𝑘〉 o
he deg ee o he ne wo ks. The au ho s also concluded ha he a o emen ioned condi ion
leads Coope a o s (De ec o s) o a ixa ion p obabili y 𝜌𝐴(𝜌𝐷), such ha 𝜌𝐴>1
𝑁>𝜌𝐷, wi h
1
𝑁 s anding o he ixa ion o a single Coope a o wi h neu al d i , i.e. w=0 in equa ion 83.
These esul s holds o la ge popula ions wi h 𝑁≫〈𝑘〉.
Real ne wo ks ha e ol ed by adding nodes, such ha new nodes a ach andomly o
exis ing nodes wi h a p obabili y p opo ional o he deg ee o he la e a e be e desc ibed
by a SF ne wo k. Aged nodes end o ha e g ea e deg ee. Along he hesis, he BA model, a
SF ne wo k wi h

=3, is adop ed. These he e ogeneous ne wo ks p o ide no only an
adequa e opology o Coope a o s o ga he , in e ac and ein o ce mu ual payo s, bu also
unc ion as a e e ence landma k o coope a ion s a egy in luence o e he neighbou hood
due o he high payo achie ed.
The impac o he e ogeneous ne wo ks on coope a ion e lec ed in he igh panel o igu e
11 is poin ed ou in (San os & Pacheco, 2005) o 2-Playe s games and la e on ex ended o
PGG in (San os, San os, & Pacheco, 2008). Ini ially De ec o s pe o m well in e ac ing wi h
57
high-deg ee nodes, bu soon a e ic ims o hei own success as hei neighbou s cease being
Coope a o s. This allows a Coope a o o in ade he hub, seed a clus e and a oid exploi a ion
by o he De ec o s.
(Gómez-Ga deñes, Campillo, Flo ía, & Mo eno, 2007) show ha a mo e he e ogeneous
deg ee dis ibu ions esul s in inc eased coope a ion by measu ing he equency o p opo ion
o playe s ha consis en ly play ei he coope a ion o de ec ion a e he sys em eaching
equilib ium. In SF ne wo ks, he e is a single la ge co e o pu e Coope a o s di icul o in ade
o ganized in hubs. The smalle numbe o hubs on andom ne wo k is he eason o he lowe
le el o coope a ion in hese ne wo ks.
2.5. OTHER RELEVANT BIBLIOGRAPHY FOR THE THESIS
I p e ious sec ions a ge ed o es ablish a common g ound o consis en e minology and
o build up he ounda ions o a heo e ical amewo k, his sec ions accoun s o bibliog aphy
mo e speci ic o he subjec s o mul ilaye ne wo ks, modelling and s uc u al measu es,
suppo o e olu iona y dynamics and he so o Public Goods Games (PGG) ha can be
played in hese s uc u es.
(San os, San os, & Pacheco, 2008) ocus on PGG in a single laye . Each node wi h k deg ee
pa icipa es in 𝑘+1 games wi h a s a opology, cen e ed on each o i s 𝑘 neighbou s and
i sel .
The only wo s a egies conside ed a e coope a ion and de ec ion. The i ness o a playe
is associa ed wi h he accumula ed payo esul ing om all PGG in which he/she pa icipa es.
S a egy e olu ion is implemen ed ia eplica o dynamics: a each ime s ep, each indi idual
adop s he s a egy o a andomly chosen neighbou wi h a p obabili y p opo ional o he
payo ( i ness) di e ence.
The wo pa ame e a iables conside ed a e he ype o ne wo k, ei he egula g aphs o
SF, and he in es men o apply pe playe in each game. Pe Coope a o playe , ei he he e
is a ixed in es men o be applied pe game, o he e is a ixed in es men pe playe o be
equally di ided among he k+1 games he/she plays. Simula ion esul s show ha ne wo k
he e ogenei y a ou s coope a ion. Addi ionally, sha ing a ixed in es men ac oss all games,
i.e., ha ing ixed in es men by playe , lowe s he syne gy ac o no malized by 𝑘+1
h eshold beyond which coope a ion becomes iable.
A ma hema ical analysis o he esul s p esen ed in (San os, San os, & Pacheco, 2008) is
de eloped by (Pacheco, Pinhei o, & San os, 2009) in o de o explain why di e si y on
con ibu ion and on ne wo k a ou coope a ion. Condi ions o a Coope a o o in ade a
58
De ec o hub a e p esen ed o bo h cases o ixed in es men pe playe o pe game.
Condi ion o la e al e na i e is less s ingen ha i s coun e pa o he o me one. As in
(San os F. C., Pinhei o, Lenae s, & Pacheco, 2012) he AGoS e lec ing he a ia ion
(inc ease o dec ease) in he numbe o Coope a o s as a unc ion o Coope a o s equency is
plo . The plo s un eils ha he unde lying s uc u e e ec i ely ans o ms a local coope a i e
dilemma in o a global coo dina ion game. Fo ixed in es men pe game wi h unde lying SF
ne wo ks, he e is a c i ical ini ial le el o coope a ion, unc ion o applicable syne gy ac o ,
below (abo e) which he le el o coope a ion ends o an All-De ec o s (Coope a o s)
ne wo k.
Mo eo e , changing he con ibu i e scheme om ixed in es men pe game o ixed
in es men pe playe in a SF ne wo k popula ion s uc u es changes a PD e ec i ely in o a
Ha mony Game whe e coope a ion become ad an ageous i espec i ely o hei
concen a ion, which means e olu ion mo ed o an ALL-Coope a o s ne wo k.
2.5.1. Mul ilaye ne wo ks
A o malism o deal wi h sys ems composed o se e al laye s, ei he wi h bina y o
weigh ed links, is p oposed by (Ba is on, Nicosia, & La o a, 2014). Di e en pe spec i es o
desc ip ion o a mul iplex ne wo k a e in oduced: he agg ega ed opological ma ix, he
o e lapping and he weigh ed o e lapping ma ix, which a e simple and mo e compac
s uc u es, bu no so ich as he adjacency ma ix 𝐴𝑖𝑗 pe laye . Basic me ics o cha ac e ize
he s uc u al p ope ies o a single laye ne wo k such as deg ee dis ibu ion, node clus e ing,
sho es pa hs, be weenness o closeness a e ex ended o a mul iplex scena io. New measu es
as mul iplex deg ee en opy eme ge. The e is a ocus on he quan i ica ion o he pa icipa ion
o single nodes o he s uc u e o each laye , and on i s impo ance o he o e all e iciency
o he mul iplex ne wo k, in e ms o node eachabili y and clus e ing. P oposed measu es a e
es ed and alida ed on a genuine mul iplex eal-wo ld da ase , he one o Top Noo din
Te o is Ne wo k (Ba is on, Nicosia, & La o a, 2014).
2.5.2. Coope a ion in Mul ilaye Ne wo ks
(Li, Wang, & Sheng, 2017) s udy in a single laye ep esen ing a 2D space he e olu ion
o coope a ion on ne wo ks ha inco po a e geog aphical cos s in o he payo unc ion o
e olu iona y games. The longe he dis ance be ween a pai o playe s, he highe he spa ial
cos inco po a ed in o he payo ma ix o he game.
Nodes ha e weigh s s anding o i s ele ance, e.g. popula ion o a ci y. Ne wo ks a e ully
connec ed wi h s uc u es geog aphically induced, as bo h nodes’ ele ance and hei dis ance

59
a e aken in o accoun in a g a i y alike unc ion, used in cos -bene i analysis when deciding
which links o inco po a e in he ne wo k.
Nodes ha e 2D coo dina es and each link is cha ac e ized by he Euclidean dis ance
be ween he nodes i connec s. PGG a e played bu he amoun in es ed by a playe in each
game is now p opo ional o a cos unc ion gi en by a Fe mi dis ibu ion applied o he
Euclidean dis ance o each o his neighbou s.
A ele an esul is ha a pola ized dis ibu ion o geog aphical cos s can signi ican ly
lowe he h eshold alue o he syne gy ac o o coope a ion in ne wo ked PGG o succeed.
In o he wo ds, he geog aphical mechanism, which pola izes coope a i e cos s is able o
lowe he h eshold alue o he syne gy ac o o coope a ion in PGG. On he o he hand,
mo e uni o m alike dis ibu ion o geog aphical cos s hinde s he e olu ion o coope a ion.
(Nakamu , Nagashim, & Yasu ak, 2015) s udy a wo laye mul iplex wi h andom
ne wo ks, small-wo ld ne wo ks, ollowing Wa s-S oga z model, and SF g aphs, based on
Ba abási-Albe model. Nodes appea in all laye s wi h no in e -laye links and ha e a single
s a egy ac oss laye s. 3 ne wo k ypes in 2 independen laye s, in a o al o 32 combina ions
a e simula ed. In each laye , PD is played aking in o accoun payo s acqui ed exclusi ely in
ha laye . Each nodes upda es i s s a egy by copying i om a neighbou wi h a p obabili y
linea ly p opo ional o hei laye payo di e ence no malized by he maximum deg ee
be ween hem. Once a node upda es i s s a egy on one laye , i assumes i in all o he laye s
so ha a node’s s a egy is cohe en ac oss laye s. In his model, coope a ion only has a chance
i bo h laye s a e SF.
(Ba is on, Ma jaz, & La o a, 2017) explo e mul iplexes wi h laye s o med by egula
andom g aphs, nodes wi h independen s a egies pe laye . Playe ’s playo is accumula ed
ac oss laye s. Fo a common syne gy ac o ac oss all laye s, he mo e o e lapping a e links
ac oss laye s, he lesse is he c i ical syne gy ac o o coope a ion o se in. Mo eo e , o a
gi en posi i e a e age node o e lapping, he bigge he numbe o laye s he smalle he
c i ical syne gy ac o o coope a ion o p e ail. In a wo laye mul iplex case, ha ing
di e en syne gy ac o s in he laye s allows new abso bing s a es wi h Full Coope a ion on
one laye and Full De ec ion on he o he .
(Li, Shen, & Jiang, 2016) ackle a scena io o playe s ha ing limi ed esou ces a ailable o
be spen in PGG played ac oss all mul iplex laye s. An agen is ep esen ed by a uple o
nodes, one pe laye . The syne gy ac o is common o all laye s. Laye s ha e homogeneous
ne wo ks, E dős–Rényi o small-wo ld models, o SF he e ogeneous models. Each agen has
60
an in es men alloca ion, an a ay o in es men s, one elemen pe laye , such ha he
in es men pe laye is non-nega i e and he o al in es men pe agen is opped. Agen
payo s a e calcula ed pe laye .
Agen s ha e a iable con ibu ions ac oss laye s wi h null con ibu ions allowed. An Agen
beha es as a Coope a o o a De ec o when his/he con ibu ion is posi i e o null,
espec i ely. Pe i e a ion, an agen is andomly selec ed. Selec ed agen payo s ac oss laye s
a e compa ed and based on Fe mi dis ibu ion, a laye is selec ed. In he chosen laye , he
ocal node adop s he alloca ion s a egy o a andom neighbou in he same laye wi h Fe mi
p obabili y dis ibu ion applied o ocal and neighbou nodes’ payo s in ha laye . In case
alloca ion s a egy is copied, u he no maliza ion o he new in es men alloca ion o he
ocal agen is equi ed in o de o keep i s o al opped.
G eedy- i s mechanism was coined o he scena io when an agen in choosing he laye
o play p e e s o upda e he alloca ion s a egy in he highe payo laye as de e mined by
Fe mi dis ibu ion.
Simula ions o 2-laye mul iplex e eal ha g eedy- i s agen s can pe o m coope a i e
beha iou s in mul iplex ne wo ks when one laye is SF ne wo k and deg ee di e ences
be ween pee nodes inc ease. An addi ional conclusion is ha deg ee di e si y and g eedy-
i s mechanism can de ea emp a ion o de ec i e beha iou s and a oid he ex emely biased
coope a ion in a ce ain laye .
A c i ic o he pape om he au ho o his hesis is ha pa s o he o malism in oduced,
in pa icula he g eedy- i s mechanism, lacked gene aliza ion o mul iplexes o mo e han 2
laye s.
(Hayashi, Suzuki, & A i a, 2016) add ess a mul iplex scena io whe e an agen canno a o d
o play PGG in mo e han 1 laye a any ins an o ime. Thus, an agen can ha e a node in any
laye , bu exclusi ely, i.e., in any poin in ime an agen will ha e a single node ha will be
posi ioned in only one laye , al hough his p esence laye can dynamically e ol e o e ime.
Coope a ion and de ec ion a e he applicable s a egies. Game e olu ion allows an agen o
change he laye whe e i s single node lays.
All laye s, independen , a e de ined as E dös–Rényi andom g aph wi h he same cons an
deg ee.
The game oughly e ol es as ollows. Fo each agen he payo o i s ep esen a i e node
is calcula ed in i s laye o p esence by playing PGG wi h i s neighbou s as dic a ed by laye
ne wo k who a e necessa ily p esen in ha laye . Wi h payo s calcula ed, o each agen a
61
po en ial neighbou , no necessa ily p esen in he same laye , is iden i ied. A e e ence alue
unc ion o node’s payo s dic a es he p obabili ies wi h which s a egy and laye o p esence
is copied om ocal agen o i s neighbou o he o he way a ound.
Wi h a bi a y p obabili ies mu a ions hi agen s: laye o p esence o i s single node
upda es o a andomly chosen new one and s a egy o i s node changes.
As he numbe o laye s inc ease, o he hings being equal, coope a ion equency
inc eases, so does he no malized en opy o he dis ibu ion o he p esence o agen s pe
laye , esul ing his inc ease om he cyclic coe olu ion p ocesses o game s a egies and
laye selec ion s a egies.
Depa ing om a scena io wi h he same E dös–Rényi andom g aph ac oss all laye s and
hen andomly ewi ing hem in each laye , i was also showed ha he e ogenei y among
laye s is a ca alys in mul iplex ne wo ks o acili a e he e olu ion o coope a ion. In a way,
his esul is a eminiscence o he esul s ob ained by (San os F. C., Pinhei o, Lenae s, &
Pacheco, 2012) in a single laye case.
(Kleinebe g & Helbing, 2018) s udy mul iplexes wi h independen SF ne wo ks. Nodes
ha e independen s a egies pe laye . E olu ion o he sys em is go e ned by imi a ion
dynamics, which means indi iduals end o adop he s a egy o mo e success ul neighbou s.
In each ound o he game, i s ly each node chooses one laye andomly and hen, wi hin his
laye , one neighbou a andom. Neighbou ’s s a egy o ha laye is copied wi h a p obabili y
gi en by Fe mi dis ibu ion applied o he di e en ial o nodes payo . The payo o a node
is calcula ed as he a e age payo accomplished ac oss all laye s.
I PD is he game played and he e is no deg ee co ela ion be ween laye s, inc easing he
numbe o laye s only leads o minimum changes in inal le el o coope a ion om a single
laye scena io. Howe e , i deg ee co ela ions a e p esen and he numbe o laye s is la ge
enough, an a e age le el o coope a ion o 0.5 domina es he whole T-S pa ame e space and
jus i ies he ‘ opological ensla emen ’ i le o he a icle. Topological ensla emen eme ges
and highligh s payo i ele ance as bo h he numbe o laye s and he s eng h o deg ee
co ela ions a e inc eased.
Topological ensla emen is also obse ed wi h he Ha mony game (𝑆=𝑇=0.5) and PD
wi h 𝑆=−0.5,𝑇=1.5. In a single laye , o me game ends o All Coope a o s; la e one o
All De ec o s. Howe e , he same games wi h a su icien la ge numbe o laye s and s ong
deg ee co ela ion among laye s con e ge o a Coope a o concen a ion o 0.5.
Topological ensla emen implies also ha he ini ial mul iplex le el o coope a ion
de e mines he inal dis ibu ion o he le el o coope a ion ac oss laye s, almos i espec i e
62
o T-S pa ame e combina ion suppo ing he game played, subjec o he condi ion o exis ing
a s ong deg ee co ela ion ac oss laye s.
Topological ensla emen is no exclusi e o 2-Playe games like he PD o he Ha mony
game. I also occu s in PGG. PGG a e played independen ly in di e en laye s, and he o al
payo s a e agg ega ed and a e aged pe node. The e is a ixed in es men pe playe pe laye .
Coope a o s’ equency ge s la ge in he absence o deg ee co ela ions, bu i his co ela ion
is p esen and he numbe o laye s inc eases, Coope a o s’ equency d ops o a ixed alue
a ound 50%. These esul s a e independen o simila i y co ela ion o o e lapping, a end
nodes may ha e o connec o he same neighbou s in di e en laye s.
69
Table 6- Lis o he Pa ame e s used in he Compu e Simula ion model
Lis o pa ame e s used
 In es men C i e ia
 Ne wo k ype, BA o Ho and, common ac oss all laye s
 Numbe o laye s (M)
 In ensi y o selec ion (), wi hin he in e al be ween 0.01 and 10.0
 Enhancemen ac o (F), wi hin he in e al be ween 1 and 2
Table 7E o ! Re e ence sou ce no ound. summa ises he me ics ob ained om
compu e simula ions used o conduc he discussion and suppo ou indings.
Table 7- Lis o all he Me ics collec ed om Compu e Simula ions
Me ics o collec om Compu e Simula ions
 Le el o Coope a ion co esponding o he a e age le el o coope a ion o e all laye s snapsho ed a
he end o each simula ion
 Dis ibu ion o he p obabili y o he numbe o Coope a o s pe laye and agg ega ed, i.e., a e aged
ac oss all laye s, as obse ed du ing simula ions pe iod
 Dis ibu ion o he p obabili y o he numbe o Coope a o s pe laye and agg ega ed as snapsho ed a
he end o each simula ion
 Le el o Coope a ion co esponding o he a e age le el o coope a ion o e all laye s snapsho ed a
he end o each simula ion
 Dis ibu ion o he p obabili y o he numbe o Coope a o s pe laye and agg ega ed, i.e., a e aged
ac oss all laye s, as obse ed du ing simula ions pe iod
 Dis ibu ion o he inal e olu iona y ou come disc imina ed pe laye
 A e age G adien o Selec ion (AGoS) pe laye and a e aged ac oss all laye s snapsho ed a he end
o egula numbe o gene a ions and a he end o each simula ion
 S a egy cohe ence o node consis ency
Dis ibu ion o he inal e olu iona y ou come pe laye is snapsho ed a he end o
simula ions in each o he ollowing disjoin condi ions: (i) All laye s sa u a ed as ALLC; (ii)
All laye sa u a ed as ALLD; (iii) All laye s sa u a ed, a leas one as ALLC and a leas one
as ALLD; and (i ) A leas one laye no sa u a ed as ALLC o as ALLD.
S a egy cohe ence o node consis ency, which is compu ed as
 = 1
𝑁∑1
𝑀
𝑁
𝑛𝑜𝑑𝑒=1 |∑𝑆∗𝑖𝑙
𝑀
𝑙𝑎𝑦𝑒𝑟 𝑙=1 |
(93)
wi h

70
𝑆∗𝑖𝑙 = {1, o a Coope a o
-1, o a De ec o
(94)
is measu ed a he end o each simula ion and a e aged ac oss all simula ions. I he e a e 𝑀
laye s and he node is Coope a o in 𝑀𝐶 o hem, hen consis ency equals |1−2𝑝|, wi h p =
𝑀𝐶/𝑀. Consis ency as a unc ion o 𝑝 is a 𝑣 line wi h a minimum o 0 a 𝑝=0.5.
3.2.1. Nume ical Me hods
Calcula ing AGoS is e y CPU demanding. Taking in o accoun ha he consequence o
an indi idual upda ing his/he s a egy is ha i s in es men s ac oss laye s a y, i s neighbou s
a dis ance 1 ac oss all laye s ha e hei payo upda ed and ha he impac on indi idual
con ibu ions o AGoS a laye le el is ci cumsc ibed o neighbou s a dis ance up o 2 in any
laye , an algo i hm was concei ed o inc emen ally upda e mul ilaye payo s and AGoS a
each s ochas ic p ocess i e a ion, ins ead o ecalcula ing hese me ic in eg ally om sc a ch.
Appendix A p esen s he algo i hm ollowed o inc emen ally calcula e AGoS and payo s
a each i e a ion o he s ochas ic p ocess ha ing nume ical op imiza ion in mind. The
algo i hm concei ed esul s om a ade-o be ween a lesse bu den on CPU ime and an
inc ease in memo y consump ion, wi h mo e auxilia y a iables being conside ed. Al hough
a ac i e because o i s pe o mance, he downside o adop ed inc emen al upda ing o
Table 8- Compa ison on he Numbe o Ope a ions equi ed o Payo and AGoS Calcula ion wi h and
wi hou Nume ical Op imiza ion
A e age numbe o Ope a ions pe a Node S a egy upda e in a Laye
Ope a ions
Unop imized Ve sion
Op imized Ve sion
# o indi idual Payo
Calcula ions
𝑁𝑀
Baseline:
2〈𝑘〉1
〈𝑘〉+1=3(1)
Dis ibu ed c i e ia:
2𝑀〈𝑘〉1
〈𝑘〉+M=3M(1)
# o indi idual AGoS
Calcula ions
𝑁𝑀
𝑀(〈𝑘〉+1)+2𝑀1
〈𝑘〉〈𝑘〉2=
𝑀(3〈𝑘〉+1) (3)
(1)F om s eps 2, 5 and 6 o he algo i m (see appendix A)
(2)F om s eps 1, 7 and 8 o he algo i m (see appendix A)
71
a iables pe i e a ion is ha ounding e o s due o machines ini e p ecision accumula e.
Measu es o mi iga e and con ol his consequence a e also men ioned in appendix A.
The compa ed numbe o ope a ions in op imized and non-op imized calcula ions o each
i e a ion o he s ochas ic p ocess on a mul ilaye wi h 𝑀 laye s o ne wo ks, each wi h 𝑁
nodes and 〈𝑘〉 a e age deg ee, is as in able 8.
In non-op imized e sion i is assumed ha a e e y i e a ion all payo s and AGoS a e
calcula ed om sc a ch.
As 𝑁≫〈𝑘〉, he sa ing o CPU u iliza ion in he op imized calculus o s ochas ic p ocess
e olu ion is e iden . We e o he ope a ions as he calculus o laye and mul ilaye agg ega ed
AGoS conside ed and he gain on he op imized e sion side would be s eng hened.
72
4. RESULTS AND DISCUSSION
The goal o his wo k is o explo e how di e en in es men c i e ia impac s he e olu ion
o coope a ion. Gi en he la ge pa ame e space, ou i s app oach was o slice i along a
numbe o dimensions. Tha means, keeping some a iables cons an while explo ing he
impac o a ying he emaining along a p ede ined domain. Hence, i made i possible o
in es iga e he impac o di e en condi ions g aphically and de i e some in ui ion on he
unde lying mechanics o coope a ion p omo ion.
Figu e 14 esul s om a coa se g ain sweeping o he pa ame e space and plo s le els o
coope a ion a e aged ac oss all laye s o he mul ilink colou coded. I is an eagle’s eye iew
on how he 5 pa ame e s conside ed, in es men c i e ia, ne wo k ype, numbe o laye s,
in ensi y o selec ion (𝛽) and enhancemen ac o (𝐹), de e mine he le els o coope a ion
a ained. In a single go i a emp s bo h o oughly assess he impac o di e en esou ce
in es men c i e ia in he e olu ion o coope a ion in popula ions in e ac ing h ough a
mul ilaye ne wo k and o assess how sensi i e is he o e all mul ilaye beha iou o
en i onmen pa ame e s, in pa icula he numbe o laye s and unde lying ne wo k ypes.
Figu e 14- Le el o Coope a ion as a Func ion o Ne wo k ype, In es men C i e ia, Numbe o Laye s,
In ensi y o Selec ion (𝛽) and Enhancemen Fac o (𝐹). Each laye wi h 𝑁=1000 indi iduals was ini ialized
wi h hal o hem andomly chosen as Coope a o s. The e is no deg ee-deg ee co ela ion o o e lapping.
O e all, we concluded ha in he p esence o dis ibu ed in es men c i e ia, le els o
coope a ion inc eased wi h inc easing numbe o laye s. In opposi ion, o baseline c i e ia he
le el o coope a ion has li le sensibili y o a ia ions in he numbe o laye s. This is because
adding mo e laye s o a mul ilaye wi h baseline c i e ia has a limi ed impac on o iginal
73
laye s. Pa ial payo o indi iduals on o iginal laye s su e s no change; only he p obabili y
o copying he s a egy om a neighbou changes as accumula ed payo pe indi idual is
upda ed. On he o he hand, wi h dis ibu ed in es men he addi ion o a laye comple ely
changes he game. By o cing a ac ion o he in es men o each indi idual o low owa ds
he new laye , as long as he indi idual coope a es in he new laye , such low o in es men
impac s on he payo s o indi iduals’ neighbou s in all laye s he coope a es. This c ea es a
new dynamics by changing bo h indi iduals’ pa ial payo s in o iginal laye s as well as he
p obabili y o copying s a egies om neighbou s.
When all o he a iables a e se as cons an , coope a ion imp o es wi h inc easing
enhancemen ac o (𝐹) as expec ed, because a g ea e F implies ha in he payo ma ix o
DPD Coope a o s’ payo s ge close o De ec o s’ ones, hus p omo ing Coope a ion
3
.
Addi ionally, BA ne wo ks lead o b oade condi ions o he p omo ion o coope a ion han
Ho and ne wo ks.
Conce ning he ole o he in ensi y o selec ion (𝛽), we obse e ha coope a ion is
a ou ed in s ong selec ion egimes wi h la ge enhancemen ac o s. When he numbe o
laye s inc eases, coope a ion becomes dominan ac oss he en i e ange o selec ion p essu es
o an in ensi y o selec ion abo e a c i ical h eshold ha aises wi h he numbe o laye s,
bu in a way ha is no ans e sal o all pa ame e scena ios and hus equi es a segmen ed
analysis. No iceable in each hea map (see igu e 14), unde dis ibu ed in es men c i e ia
and o inc easing numbe o laye s he e is a egion on he le side wi h an uni o m colou
co esponding o hal way he colou map conside ed, p ecisely he same le el o coope a ion
wi h which laye s we e ini ialized. No iceable also is ha his egion ge s la ge wi h i s igh
on ie mo ing igh wa ds as he numbe o laye s inc eases. Wi hin his egion, he
coope a ion le el ends o become insensi i e o enhancemen ac o (𝐹). The o ma ion o
his egion is mo e e iden in Ho and ne wo ks and wi h in es men dis ibu ed pe laye
c i e ia, bu al hough no expe imen ed in he pa ame e subdomain unde lying he igu e i is
ex ensible o he o he dis ibu ed in es men c i e ia and BA ne wo ks. The phenomenon in
place is a combined opological and c i e ia ensla emen ha u he on we will elabo a e on
and ha in sho boos s mul ilaye ine ia o change, leading he mul ilaye o p ese e
3
In each i e a ion, would he choice o he laye whe e o play he DPD game be
de e minis ic, e.g. by choosing he laye whe e an indi idual maximizes i s local payo , and
coope a ion le el would no inc ease wi h enhancemen ac o .
74
h oughou simula ions wha e e le el o coope a ion i was ini ialized wi h. Cha ac e is ic o
his ensla emen egion is also ha he maximum in ensi y o selec ion (𝛽) delimi ing i s
bo de a ies in he same di ec ion as he numbe o laye s.
Unde he baseline c i e ia and wi h ou BA laye s o mo e, coope a ion le els a e high
(a ound 80%) in s ong selec ion and high enhancemen alues (F). This end has a di e en
mo i a ion han he one poin ed ou o dis ibu ed in es men c i e ia cases and had al eady
been iden i ied in (Kleinebe g & Helbing, 2018). Wha happens is ha in he absence o
co ela ions be ween laye s as hei numbe inc eases so do he odds o an indi idual being a
Coope a o and a hub in a leas one o he laye s. As payo s a e accumula ed and new laye s
do no change in es men al eady applied, he con ibu ion om new coope a i e hubs helps
an indi idual bo h o hold his/he g ound in laye s whe e he/she plays coope a ion wi h ew
neighbou s as in laye s whe e de ec ion was he op ion. We conside ed F = 1.7 in baseline
c i e ia, and we obse ed ha a e age coope a ion le el ops a a ound 80% wi h he
mul ilaye in a s able equilib ium poin wi h laye s holding indi iduals wi h bo h s a egies
and indi iduals wi h dis inc s a egies ac oss laye s. Figu e 15 plo s he p obabili y o inding
an 8-laye baseline mul ilaye a a pa icula a e age le el o coope a ion by he end o a
simula ion. The expec ed alue o his dis ibu ion co esponds p ecisely o he alues
measu ed in igu e 14.
Figu e 15- Quasi-S a iona y P obabili y o an 8-Laye Mul ilaye wi h BA Ne wo ks and Baseline
In es men C i e ia. In ensi y o Selec ion (𝛽) alues 1 and he numbe o indi iduals (𝑁) pe laye equals o
1000. P obabili ies we e calcula ed a he end o simula ions. Each laye was ini ialized wi h hal indi iduals
andomly chosen as Coope a o s. The e is no deg ee-deg ee co ela ion o o e lapping.

75
In o de o sh ink he pa ame e space and zoom in in o di e en sub-domains whe e
coope a ion is expe ienced, e e ence alues we e iden i ied o he in ensi y o selec ion (

).
Values chosen we e 1.0 and 0.1, based on he a ional ha mul ilaye s wi h his
pa ame iza ion can span he en i e ange o coope a ion le els.
Figu e 16 explo es how he le el o coope a ion esponds o di e en slices in pa ame e
space de ined by an in ensi y o selec ion (

); enhancemen ac o (𝐹) and numbe o laye s
(M). In each column we conside scena ios whe e wo pa ame e s a e cons an and he
emaining ones a y wi hin he in e al o analysis.
Figu e 16- Le el o Coope a ion in Mul ilaye Ne wo ks in 2-Playe Dis ibu ed P isone Dilemma (=0.1).
Le el o coope a ion was measu ed and a e aged a he end o simula ions. Ne wo ks in each laye ha e 1000
indi iduals (𝑁), hal andom and independen ly ini ialized as Coope a o s.
The uppe and bo om le panels show he le el o coope a ion as a unc ion o
enhancemen ac o (𝐹), wi h he numbe o laye s se o 8 and in ensi y o selec ion 𝛽=0.1.
Fo BA ne wo ks, unde he baseline c i e ia o F lowe han 1.5 we obse e ha ull de ec ion
(ALLD) is he dominan ou come, and o F g ea e han 1.8, popula ion eaches a ull
coope a ion s a e (ALLC). In be ween hese alues, we wi ness indi iduals changing s a egy
om De ec ion o Coope a ion. The exis ence o a c i ical enhancemen ac o (𝐹) beyond
which coope a ion le els ise had al eady been iden i ied in (Pacheco, Pinhei o, & San os,
2009) o a single laye . Al hough no shown, esul s om his a icle we e eco e ed.
In es men dis ibu ed pe laye line has a simila end o he one conce ning baseline. I
in es men is dis ibu ed pe game coope a ion aises s eadily wi h 𝐹, bu wi h a low slope.
76
Unde he baseline c i e ia wi h Ho and ne wo ks he c i ical enhancemen ac o (𝐹) is
loca ed nea 𝐹=1.9, meaning ha i is a mo e s ingen con ex o he e olu ion o
coope a ion. Only beyond his alue, he e is oom o coope a ion. Unde he dis ibu ed
in es men c i e ia he le el o coope a ion inc eases mono onically wi h F un il eaching a
le el o coope a ion o 0.5 o 𝐹=2.0. This beha iou is due o hese slices cu ing he
pa ame e space almos en i ely wi hin he a o emen ioned egions o uni o m coope a ion
le el o 0.5 in igu e 14.
The middle op and bo om panels show he e olu ion o coope a ion while a ying numbe
o laye s in he mul ilaye . We conside a cons an enhancemen ac o o 1.7 and an in ensi y
o selec ion o 𝛽=0.1. The selec ion o he enhancemen ac o ook in o conside a ion he
loca ion o he le el o coope a ion phase ansi ion obse ed on BA on op le panel.
On BA ne wo ks (uppe middle) panel we obse e he exis ence o an op imum numbe o
laye s ha maximizes coope a ion. This maximum can be explained by he combina ion o
wo mechanisms wi h opposing e ec s: i s , as he numbe o laye s inc eases he c i ical F
beyond which coope a ion is igge ed dec eases, as he case o in es men pe laye c i e ia
is he mos e iden one; secondly, as he numbe o laye s inc eases he coope a ion le el
ends o each a ixed alue de e mined by he ini ial p opo ion o coope a ion, a phenomenon
known as ensla emen egion (see appendix C). The baseline ela ed cu e aises wi h he
numbe o laye s bu opped by he al eady men ioned ba ie a ound 80%.
On Ho and ne wo ks and in he bo om middle panel we explo e a scena io whe e he
enhancemen ac o (𝐹) lies in a egion o ull de ec ion. Hence, a simila ou come is ob ained
when we inc ease he numbe o laye s. Howe e , when in es men is dis ibu ed pe laye ,
coope a ion becomes easible in ne wo ks wi h ou o mo e laye s. Fo in es men dis ibu ed
pe game, coope a ion becomes easible o ne wo ks wi h wo laye s o mo e. Unde
dis ibu ed in es men c i e ia, he e is hus a c i ical numbe o laye s o coope a ion o
eme ge.
Rele an om middle panels is also ha o bo h dis ibu ed in es men c i e ia and o
bo h ypes o ne wo ks, al hough mo e e iden in he Ho and case, he le el o coope a ion
con e ges o 0.5 as he numbe o laye s inc ease. Again, his is due o poin {𝛽,𝐹}=
[0.1,1.7} o he numbe o laye s sampled lying wi hin he a o emen ioned egions o
uni o m le el o coope a ion o 0.5 in igu e 14.
Finally, on he uppe and bo om igh panels we explo e he le el o coope a ion as a
unc ion o he in ensi y o selec ion (

), o ne wo ks wi h 8 laye s and an enhancemen ac o
o 1.7. On BA ne wo ks, coope a ion is sus ained in he en i e in e al o selec ion p essu es
77
in es iga ed. Mo eo e , he e is an op imal in ensi y o selec ion (

) ha maximizes he le el
o coope a ion. A inding ha has been obse ed p e iously in single laye ed ne wo ks
(Pinhei o, Pacheco, & San os, 2012).
On Ho and ne wo ks, since o F = 1.7 unde he baseline c i e ia a egime o ull de ec ion
is in place, he e is li le impac in a ying he in ensi y o selec ion (

). Howe e , o he
dis ibu ed in es men c i e ia, we obse e signi ican gains in he le el o coope a ion o
inc easing selec ion p essu e and in pa icula when in es men s a e dis ibu ed pe laye .
Fo bo h dis ibu ed in es men c i e ia and bo h ne wo ks, igh panels also highligh he
ac ha o minimal in ensi y o selec ion (

), he le el o coope a ion eached coincides wi h
he one o mul ilaye ini ializa ion. The de ia ion inc eases as in ensi y o selec ion (

) aises.
As expe imen ally e i ied, we e he igu e eplica ed wi h a highe numbe o laye s and he
c i ical

un il which ini ial and inal le els o coope a ion a e con e gen would inc ease.
Al hough no shown he e, we ha e epea ed he case whe e he i ness o indi iduals
co esponded o he a e aged payo ac oss laye s, ha is, ins ead o he accumula ed payo .
In ha con ex , he esul s a e simila o he ones discussed in he e wi h a escaled selec ion
p essu e o accoun o he no maliza ion done by he numbe o laye s.
Mo eo e , in appendix B we explo e he impac o deg ee-deg ee co ela ions in mul ilaye
social ne wo ks in he e olu ion o coope a ion o he h ee in es men c i e ia unde analysis.
We show ha while inc easing deg ee-deg ee co ela ions widens he ange o dilemmas o
which coope a ion is p e alen , he egion o pa ame e s whe e he popula ion is able o each
an ALLC s a e dec eases subs an ially.
4.1. TOPOLOGICAL ENSLAVEMENT UNDER DISTRIBUTED INVESTMENTS AND
LARGE NUMBER OF LAYERS
I is impo an o de ail he limi ing case o opological ensla emen ha eme ges when he
numbe o laye s is e y la ge. This scena io is speci ic o dis ibu ed in es men c i e ia,
al hough i can also be obse ed unde he baseline c i e ia in es ic ed subdomains o he
pa ame e space, e en ually equi ing addi ional condi ions on he mul ilaye deg ee-deg ee
co ela ion.
The p e alence o opological ensla emen s ems om he ac ha , in opposi ion o
baseline c i e ia, in dis ibu ed in es men c i e ia when an indi idual becomes a De ec o in
one laye he/she will edis ibu e his/he in es men owa ds he o he laye s whe e he/she
emains a Coope a o . As a esul , all indi iduals will ha e he same expec ed accumula ed
payo wi h minimal a iance. F om his, i esul s ha in he limi o many laye s he
78
e olu iona y dynamics eco e s a andom walk pa e n, whe e he ou come is a unc ion o
he ini ial abundance o Coope a o s and De ec o s.
Quasi-S a iona y Dis ibu ion o Ho and Mul ilaye a e aged ac oss Laye s (=
𝟏,〈𝐤〉=𝟒,𝐅=𝟏.𝟕)
Quasi-S a iona y Dis ibu ion o BA Mul ilaye a e aged ac oss Laye s (=
𝟎.𝟏,〈𝐤〉=𝟒,𝐅=𝟏.𝟕)
Figu e 17- Expe imen al Quasi-S a iona y Dis ibu ion o Mul ilaye s a e aged ac oss Ne wo k Laye s.
Laye s wi h 1000 indi iduals (𝑁) a e unco ela ed and we e independen ly ini ialized, each one wi h a numbe
o Coope a o s de e mined by a uni o mly dis ibu ed a iable wi h alues om 0 o 1000, inclusi e
85
This sec ion shed ligh on how he mic oscopic dynamics locally de ined a indi idual le el
ela es o he esul ing global dynamics and allowed o conclude on he inexis ence o pa e ns
o “sel -simila i y” a di e en scales.
4.3. AGGREGATED AVERAGE GRADIENT OF SELECTION
Up o now, we ha e analysed he e olu iona y dynamics o dis ibu ed in es men s in
mul ilaye ed social ne wo ks on he a ained le el o coope a ion. We now in oduce he
Agg ega ed G adien o Selec ion (AGoS) ool in o de o ob ain a mo e accu a e desc ip ion
o he unde lying dynamics ha leads o he obse ed ou comes. Hence, we aim a answe ing
wha is he popula ion-wide dynamics ha cha ac e izes he social-dilemma aced by he
popula ion.
In a single laye wi h ini e popula ion, he AGoS is calcula ed as he di e ence in he
p obabili y o inc ease o he numbe o Coope a o s by one and he p obabili y o dec ease
he numbe o Coope a o s by one. The AGoS, which mus be compu ed nume ically, aims o
es ima e hese wo quan i ies o all possible s a e ansi ions h ough a la ge numbe o
compu e simula ions o he e olu iona y p ocess. The AGoS is, by de ini ion, ne wo k
dependen bu con ex independen , as i eco e s he popula ion mean- ield cha ac e . The
AGoS cap u es dynamical in o ma ion equi alen o ha p o ided by eplica o equa ion in
EGT o in ini e well-mixed popula ion p esen ed in 2.1.4.
Figu e 22- A e aged Agg ega ed G adien o Selec ion (AGoS) a e aged ac oss Laye s and Time. All
ne wo ks ha e 1000 indi iduals (𝑁) wi h 〈𝑘〉=4,𝛽=1. Laye s we e independen ly ini ialized wi h a numbe
o Coope a o s gi en by a andom a iable uni o mly dis ibu ed be ween 0 and 1000, inclusi e

86
In o de o ex end he AGoS o mul ilaye social ne wo ks, we op ed o compu e he AGoS
independen ly pe laye , which is hen a e aged o ob ain a popula ion and laye wide AGoS
ep esen a ion o he dynamics a hand.
Figu e 22 shows he a e age AGoS on mul ilaye s wi h di e en numbe o laye s and
ypes o ne wo ks. We conside an in ensi y o selec ion o 1.0 (

=1.0) o bo h ypes o
ne wo ks. The c i e ia o selec ing he enhancemen ac o (𝐹) ollows om he ou comes
shown in igu e 14 o bo h ne wo k opologies and in ol es selec ing an enhancemen ac o
ha is p ecisely a he ansi ion be ween ALLD and ALLC ou comes. In ha sense, we will
be using 𝐹=1.6 o BA ne wo ks and 𝐹=1.9 o Ho and ne wo ks.
Fo baseline c i e ia and BA ne wo ks, he mul ilaye beha iou is one o coo dina ion. In
ha case, he e olu iona y dynamics is cha ac e ized by an uns able ixed poin . F om i esul s
ha depending on he ini ial abundance o Coope a o s and he loca ion o he in e nal ixed
poin all laye s will be d i en owa ds an ALLC o an ALLD ou come. In addi ion, he loca ion
o he ixed poin is seemingly in a ian wi h he numbe o laye s. Would he enhancemen
ac o (𝐹) inc ease (dec ease) he ixed poin would mo e o he le ( igh ). The BA ype o
ne wo k is esponsible o ans o ming a de ec ion dominan DPD game in a well-mixed
popula ion in o a game ha is commonly associa ed wi h a mild social dilemma o
coope a ion, a coo dina ion game o en known as a S ag Hun .
Coope a ion in Ho and mul ilaye s unde he baseline c i e ia is doomed o a small numbe
o laye s. Howe e , o 8 laye s popula ion e olu ion is such ha he possibili y o a co-
exis ence scena io wi h a s able equilib ium oo eme ges. The mul ilaye join ly wi h he
baseline c i e ia ans o m a de ec ion dominan DPD game in a well-mixed popula ion in o a
game wi h an AGoS ypical o a co-exis ence game. This means ha e en ough indi iduals
a e locally engaged in a DPD, mul ilaye leads o he eme gence o a global popula ion wide
dynamics ha p omo es he co-exis ence o Coope a o s and De ec o s.
In he case o a single laye and baseline c i e ia, esul s a e aligned wi h hose ob ained in
(Pinhei o, Pacheco, & San os, 2012). Fo in es men dis ibu ed pe laye and BA ne wo ks,
he cu es ha e a simila shape o he baseline case bu mo e le wa ds and cease sha ing he
same equilib ium poin . Wi h inc easing numbe o laye s, he ixed poin mo es owa ds 0.
Mo eo e , in a case in which he numbe o laye s is e y high he AGoS end o be ully
posi i e, ypical o a sys em e ol ing o ull coope a ion. Wi h Ho and ne wo k and a small
numbe o laye s, all de ec ion is mul ilaye a e as wi h baseline c i e ia. Howe e , as he
numbe o laye s inc ease, a s able equilib ium poin eme ges, and he absolu e alue o he
87
a e aged AGoS ends o ze o. In ac , his educ ion o AGoS magni ude was al eady
expe ienced in he BA case.
Wi h in es men dis ibu ed pe game and BA ne wo ks mul ilaye e ol es o ull
coope a ion. I is also no iceable he dec ease o he a e aged AGoS magni ude as he numbe
o laye s inc eases. In he case o Ho and ne wo ks, a simila beha iou is obse ed wi h he
excep ion ha e olu ion is di ec ed owa ds ull de ec ion in he cases ha ha e been s udied.
Common o bo h dis ibu ed in es men c i e ia is ha he magni ude o he a e age AGoS
dec eases as he numbe o laye s inc eases, some hing ha had al eady been an icipa ed in
p e ious sec ion. Ha ing AGoS wi h magni udes close o ze o simply means ha mul ilaye
end is o p ese e he le el o coope a ion wi h which i was ini ialized. In o he wo ds,
e olu ion app oaches neu al d i . Fo dis ibu ed in es men c i e ia and Ho and ne wo ks,
appendix D p esen s a ma hema ical explana ion o why AGoS magni ude ends o ze o as he
numbe o laye s inc eases.
Figu e 22 shows esul s o he a e age AGoS, ha is he a e age AGoS o all laye s
compu ed independen ly. Howe e , as laye s a e s a is ically indis inguishable among
hemsel es, indi idual AGoS conce ning each laye a e e y close be ween hemsel es and
hus close o hei a e age AGoS. Figu e 23 illus a es his esul o mul ilaye s wi h 4 laye s
and on bo h ypes o ne wo ks opologies.
Figu e 23- Agg ega ed G adien o Selec ion (AGoS) o 4 Laye s Mul ilaye o e Time. All ne wo ks ha e
1000 indi iduals (𝑁) wi h 〈𝑘〉=4,𝛽=1. Laye s we e independen ly ini ialized wi h a numbe o
Coope a o s gi en by a andom a iable uni o mly dis ibu ed be ween 0 and 1000, inclusi e.
Mo eo e , he compu ed AGoS p esen ed so a esul s om a ime a e age ha spans o e
150 gene a ions. This disposi ion has he d awback o no cap u ing he e en ual e olu ion
o e ime ha AGoS may expe ience, hus ails o cap u e he sel -o ganizing p ocess ha
88
occu s as s a egy asso men s in he ne wo ks build up. In o de o epo he AGoS empo al
dynamics, he ime in e al was ac ioned in 150 in e als o 1 gene a ion each. In each o
hese in e als, AGoS was calcula ed independen ly.
BA Mul ilaye s wi h 𝜷=𝟏,𝑭=𝟏.𝟔,〈𝒌〉=𝟒
Ho and Mul ilaye s wi h 𝛃=𝟏,𝐅=𝟏.𝟗,〈𝐤〉=𝟒
Figu e 24- Agg ega ed G adien o Selec ion (AGoS) o Mul ilaye s a e aged o e Time. Laye s wi h
1000 indi iduals (𝑁) we e independen ly ini ialized wi h a numbe o Coope a o s gi en by a andom a iable
uni o mly dis ibu ed be ween 0 and 1000, inclusi e. The noise inc eases wi h gene a ion index due o
gene a ions ha ing ewe samples. We e he numbe o 300.000 simula ions inc eased and his noise would be
educed.
Figu e 24 exhibi s snapsho s o he AGoS a di e en gene a ions o mul ilaye s wi h
di e en ne wo ks and wi h di e en numbe o laye s. A common obse a ion o all
mul ilaye s conside ed is ha du ing he i s gene a ions AGoS is always nega i e, which
means ha mul ilaye s in i s e olu iona y s eps wi ness an inc ease in he numbe o
De ec o s. G adually in subsequen gene a ions AGoS changes and esul s in he eme gence
o a basin o a ac ion ha a ou s coope a ion. The ini ial inc ease in he numbe o
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De ec o s esul s om wo mo i es. Fi s one is ela ed o he ac o Coope a o s and
De ec o s in he ini ial con igu a ion o he mul ilaye a e andomly posi ioned in he ne wo k
making he clus e ing o nodes wi h he same s a egy ex emely imp obable. Wi h DPD being
played locally, an a guably di icul game o he p omo ion o coope a ion, he AGoS
becomes nega i e. Secondly, may be simply ha in he mul ilaye coope a ion is doomed. As
popula ion e ol e and Coope a o s is adop ed ia he in es men hey a e willing o apply, we
wi ness asso men o s a egies wi h he o ma ion o hubs o coope a ion.
Fo baseline s a egy and BA ne wo ks, AGoS eco e s he uns able ixed poin as
gene a ions e ol e. A ound he 10 h gene a ion, he ixed poin eme ges o le els o
coope a ion close o 1. As gene a ions un old, he ixed poin mo es o lowe le els o
coope a ion (dec easing he coo dina ion h eshold) un il s abilizing a a alue wi h li le
sensibili y o he numbe o laye s. A simila beha iou is depic ed on Ho and ne wo ks.
He e he ixed poin is s able, hus p omo ing a co-exis ence, and s a s eme ging wi h
le els o coope a ion close o 0. As gene a ions un old he ixed poin mo es o highe le els
o coope a ion un il i s abilizes.
Fo in es men pe laye c i e ia and BA ne wo ks, AGoS also eco e s om nega i e
alues o acqui e an uns able ixed poin ha ini ially also a ises close o ull coope a ion. The
di e ence is ha his poin mo es o lowe le els o coope a ion as e as gene a ions p og ess
and wi h g ea e numbe o laye s. Wi h in es men pe laye in Ho and ne wo ks and om
ou laye s onwa ds an AGoS cha ac e ized by a co-exis ence poin eme ges. Fo a gi en
numbe o laye s he co-exis ence oo mo es owa ds la ge le els o coope a ion o la e
gene a ions. Addi ionally, no only he di e ence in co-exis ence oo be ween he same
gene a ions sh inks as he numbe o laye s inc eases as he ixed poin mo es close o ull
coope a ion.
Fo in es men pe game c i e ia and BA ne wo ks, apa om i s gene a ions, all
subsequen ones ha e posi i e AGoS. As he numbe o laye s inc eases, AGoS magni udes
o he same gene a ions dec ease. A simila beha iou is obse ed on Ho and ne wo ks,
whe e he AGoS is nega i e o all gene a ions. As he numbe o laye s inc eases, he
magni ude o AGoS educes d as ically. Fo 8 laye s, AGoS beyond he 10 h gene a ion look
like whi e noise wi h minimum magni ude.
The simple quali a i e compa ison o successi e gene a ions o AGoS conce ning he same
in es men c i e ia and ne wo k combina ion may be misleading unless we ake in o accoun
hei ela i e magni udes. Fo ins ance, o one laye he AGoS magni ude o he i s
gene a ions is much highe han o subsequen ones. I o a gi en in es men c i e ia and
90
numbe o laye s combina ion all gene a ion dependen AGoS a e a e aged, we eco e he
AGoS p esen ed in igu e 22.
He e, we ha e ocused only in mul ilaye s buil om a single ype o ne wo k. Appendix
E add esses he case o a mul ilaye wi h he e ogeneous laye s: hal as Ho and, ano he hal
as BA.

91
5. CONCLUSIONS
In his hesis we ha e used me hods om E olu iona y Game Theo y o s udy he e olu ion
o coope a ion in mul ilaye s made up o dis inc and opologically independen ne wo ks o
a single ype, be i homogeneous o he e ogeneous. Game dynamics we e modelled a
mic oscopic agen le el ia he Fe mi-upda e dynamics and explo ed h ough compu e
simula ions. In each laye , indi iduals engaged in he simple Public Goods Games in he o m
o he Dis ibu ed P isone Dilemma.
In insic o Public Goods Games is a c i e ia on how much o in es in a game an indi idual
pa icipa es as a Coope a o . Th ee c i e ia we e explo ed. One scena io o uncons ained
esou ces, he baseline one, conside ed ha indi iduals con ibu ed a ixed amoun o 1 uni
pe game, i espec i e o he numbe o dis inc pee s hey may in e ac wi h. A second one
had in es men dis ibu ed pe laye . Tha is, indi iduals had o dis ibu e he same uni o
in es men equally h ough all laye s whe e he/she pa icipa ed as a Coope a o . Thi d c i e ia
o in es men was dis ibu ed pe game, meaning ha unde ha c i e ia indi iduals had o
spli equally a ixed in es men ac oss all he games hey pa icipa ed as Coope a o s on all
laye s.
Fo he baseline in es men c i e ia, we ha e shown ha i has li le sensibili y o a ia ions
in he numbe o laye s. Fo homogenous ne wo ks o Ho and ype, he pa ame e domain
whe e coope a ion is possible has a minimum enla gemen wi h he numbe o laye s
inc easing. Adding laye s jus inc eases he size o egula space, wi hou b eaking i s
symme y, as new laye s a e s a is ically iden ical o p e ious ones wi h no impac on he
in es men s al eady applied. I he ne wo ks a e he e ogeneous o BA ype, he ones ha be e
map social ne wo ks, a sligh deg ada ion in coope a ion le els is e en expe ienced when he
numbe o laye s inc eases wi h minimum change o he domain whe e coope a ion is iable.
Fo dis ibu ed in es men c i e ia, as he numbe o laye s inc eases, we see he eme gence
o coope a ion o lowe le els o selec ion p essu e (

). In he case o in es men pe laye
and as he numbe o laye s inc eases, he p omo ion o coope a ion is mo e consis en in he
homogeneous case han in he e ogeneous one. Mo eo e , he luc ua ions o coope a ion le el
in he pa ame e domain a e smalle in he homogeneous case. Fo in es men pe game,
he e ogeneous ne wo ks a ou coope a ion o any numbe o laye s. Wi h homogeneous
ne wo ks, a highe numbe o laye s is equi ed o Coope a ion o succeed.
Simila o bo h dis ibu ed in es men c i e ia is ha as ei he he numbe o laye s
inc eases o he in ensi y o selec ion (

) dec eases, a opological and c i e ia ensla emen
92
eme ges, causing inal le el o coope a ion o coincide wi h he ini ial p opo ion o
Coope a o s, i espec i e o he alues enhancemen ac o (𝐹) en i onmen . In he pa ame e
space o he numbe o laye s e sus in ensi y o selec ion (

), a concep ual line wi h hese
pa ame e s a ying in he di ec a io can be aced such ha o any combina ion o hese
pa ame e s below he h eshold de ined by he line ensla emen ules.
Mo eo e , in he dis ibu ed in es men c i e ia, as ime un olds laye s sa u a e becoming
ei he ALLC o ALLD o he e ogeneous ne wo ks and a laye pola iza ion is eached. Thus,
he ini ial p opo ion o Coope a o s e lec s in he p opo ion o laye s ha sa u a e as ALLC.
Fo homogeneous cases, we ind a mix o pola ized laye s and pola ized popula ions, i.e.,
laye s whe e bo h s a egies coexis . Howe e , as he numbe o laye s inc eases he numbe
o laye s wi h pola ized popula ions ades away un il comple e disappea ance. Because he
numbe o laye s is disc e e, when i inc eases also dec eases e o g ain o di e ence
be ween he p opo ion o laye s sa u a ing as ALLC and ini ial Coope a o concen a ion.
This laye pola iza ion is o no su p ise, as he inc easing numbe o laye s leads he
e olu iona y dynamics o con e ge o neu al d i ha ing wo abso bing s a es, ALLC o
ALLD, wi h complemen a y p obabili ies o being eached, he p obabili y assigned o o me
s a e esul ing om he le el o coope a ion he mul ilaye was ini ialized wi h.
By de ini ion o homogeneous ne wo ks coope a ion le els a ained wi h in es men
dis ibu ed pe game coincide wi h he ones ob ained wi h in es men dis ibu ed pe game
wi h an highe in ensi y o selec ion (

) ac o . This beha iou is also expe ienced wi h
he e ogeneous ne wo ks.
Ano he challenge a ge ed was o unde s and mul ilaye global dynamics and how hey
ela e wi h mic oscopic dynamics locally de ined a agen le el. To ackle his, we eso ed o
AGoS, a nume ically compu ed measu e ha is ne wo k dependen . Ini ially concei ed o a
single laye , i was he e ex ended o he case o mul ilaye ed social ne wo ks. The AGoS
measu es he balance o p obabili ies o inc ease and dec ease he numbe o Coope a o s by
one a a gi en s a egy con igu a ion.
Fi s ly, he single laye concep was gene alized o a mul ilaye esul ing in an a e aged
AGoS ac oss laye s. Wha a e aged AGoS ac oss laye s demons a es is ha o
homogeneous ne wo ks, ega dless o he in es men c i e ia, as he numbe o laye s
inc eases he magni ude o AGoS ends o ze o (i.e., o neu al selec ion). This end is e en
emphasized o dis ibu ed in es men , which explains why he p opo ion o Coope a o s
wi h which a mul ilaye is ini ialized is p ese ed. S ill o dis ibu ed in es men c i e ia,
93
AGoS also shows ha as he numbe o laye s inc eases, al hough wi h a magni ude ending
o ze o, AGoS can become posi i e wi h an o e all dynamics o a co-exis ence scena io.
Fo he e ogeneous ne wo ks, he magni ude o AGoS also ends o ze o as he numbe o
laye s inc eases. Fo baseline and in es men pe laye c i e ia, global dynamics is one o
coo dina ion. In he o me case he oo p opo ion o Coope a o s is insensible o he numbe
he laye s whe eas in la e case i mo es le wa ds ending o ze o. A signal o wha is o
come o in es men dis ibu ed by game wi h an highe numbe o laye s is al eady e i ied
o in es men dis ibu ed pe game wi h AGoS always posi i e and ending o ze o as he
numbe o laye s inc eases.
An immedia e akeaway om his hesis is ha wi h dis ibu ed in es men a ailable,
which can be ime o sha e among many social ne wo ks o in e es , in scena ios as he ones
explo ed along he s udy de eloped, when he numbe o laye s inc eases indi iduals end o
synch onize hei s a egies and coope a e (de ec ) in he same laye s.
Ha ing his hesis been mos ly heo e ical in cha ac e , wi h due humili y and compa isons
apa , he au ho eminds he anecdo ic Ha dy example. Ha dy was an English ma hema ician
and paci is who li ed du ing wo p e ious cen u ies. He claimed o ha e ne e done any hing
use ul du ing his li e. By use ul he mean applied, ha could be used by he a my. Because he
insis ed on only wo king on pu e and abs ac ma hema ics, he ne e d eamed o i , bu much
o his wo k was la e on applied in a ious b anches o science as e.g. popula ion gene ics.
5.1. FUTURE WORK
As o sugges ions o con inuing his wo k, a numbe o di ec ions can be an icipa ed. The
dependency o coope a ion le els on a e age ne wo k deg ee was no expe imen ed, because
al hough ying di e en ypes o ne wo ks, all ne wo k ins ances sha ed he same a e age
deg ee. When inc easing 〈𝑘〉 no signi ican changes in he esul s a e expec ed, al hough
acco ding o equa ion 15 in appendix C opological ensla emen may equi e an highe
numbe o laye s o eme ge as 𝜎𝐶2
2∝〈𝑘〉, whe e 𝜎𝐶2
2 accoun s o he a iance o an indi idual
payo om neighbou s in es men . In each mul ilaye expe imen ed all laye s had he same
a e age deg ee and ype o ne wo k, no asso i eness. I would be in e es ing o b eak his
symme y in h ee independen di ec ions: cease o ha e he same a e age deg ee ac oss all
laye s o he same mul ilaye , allow di e en ypes o ne wo ks in di e en laye s as i is
pe o med in appendix E and a y asso i eness. A majo impac is expec ed on calcula ing
AGoS in scena ios whe e symme y ac oss laye s is b oken. In e ms o cha ac e iza ion,
laye s sha e no mo e a numbe o indica o s o s a is ical dis ibu ions, e.g. 〈𝑘〉, 𝑃𝑟𝑜𝑏(𝑘),
94
asso i eness, e c., which ce ainly implies AGoS pe laye o di e . The challenge is now i
and how pa ial laye s AGoS could be combined and summa ized in a single one explaining
he e olu ion o coope a ion. i.e., i he e is a cohe en game played ac oss all laye s o in
opposi ion i di e en games a e played in di e en laye s such ha a summa y AGoS ends
up e lec ing an “a e age” game played nowhe e.
In e es ing o ind ou is also whe he asymme y be ween laye s in a mul ilaye can b eak
opological and c i e ia ensla emen .
A simple ex ension o Public Goods Games and o i s Dis ibu ed P isone Dilemma
e sion would be o add ess olun a y pa icipa ion, i.e., o allow each playe o adop a hi d
s a egy in each in e ac ion, he one o Lone . When an indi idual abs ains om playing by
deciding o be Lone , his/he pee in pai wise in e ac ion is compelled o also ac as Lone and
bo h playe s a e ewa ded by an amoun 𝜎, posi i e bu smalle han he ypical alue
co esponding o a coope a ion in e ac ion. Because a Lone is be e o han a non-Lone
playing agains a De ec o , al hough wi h a smalle ewa d han i playing agains a
Coope a o , Lone is an a ac i e s a egy o he isk a e se playe s.
Di e en se s o s ochas ic ules can be concei ed o an indi idual in a laye o become o
cease beha ing as a Lone . In he limi when 𝜎=0, whene e an indi idual ac s as a Lone
in a laye , i is as his/he ela ionships we e empo a ily e ased om he laye a leas du ing
he ime in e al he/she insis s on playing as Lone . An al e na i e o he game ha ing 3
indi idual s a egies, would be o playe s o adhe e o one o wo mixed s a egies, Lone
plus a base s a egy o Coope a ion o De ec ion. Game pa icipa ion would be p obabilis ic.
101
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102
Appendix A. Algo i hm o Payo and AGoS Calcula ion
The nume ic op imiza ion conside ed in o de o make he simula ions less demanding on
CPU equi es mo e a iables (memo y) as a ade-o . The ele an se o a iables conside ed
o he op imiza ion o he simula ions’ du a ion is as ollows:
Va iable
Meaning
𝑛
-
Chosen node which s a egy is o be upda ed in…
𝑙𝑛
-
… laye 𝑙𝑛
F
-
Enhancemen ac o
M
-
Numbe o laye s
𝛽
-
In ensi y o selec ion
𝑁𝑙
-
Numbe o nodes in laye 𝑙 wi h 𝑁𝑙=𝑁
𝑆𝑖𝑙
-
S a egy o node 𝑖 in laye 𝑙
I can alue 𝐶 (Coope a o ) o 𝐷 (De ec o )
In es 𝑖𝑙
-
In es men o node 𝑖 in laye 𝑙 pe game
Payo 𝑖𝑙
-
Payo collec ed by node 𝑖 in laye 𝑙
Payo 𝑖
-
Accumula ed payo o node 𝑖 ac oss all laye s
Payo 𝑖=∑Payo 𝑖𝑙
𝑀
𝑙=1
Neigh1𝑖
𝑙
-
Se o nodes wi h di ec links o node 𝑖 in laye 𝑙
The dis ance be ween nodes 𝑖 and 𝑥 in laye 𝑙 wi h 𝑥∈ 𝑁𝑒𝑖𝑔ℎ1𝑖
𝑙
alues 1.
Neigh2𝑖
𝑙
-
Se de ined as {𝑥:∃𝑗∈Neigh1𝑖
𝑙,𝑥∈𝑁𝑒𝑖𝑔ℎ1𝑗
𝑙,𝑥≠𝑖,𝑥∉Neigh1𝑖
𝑙}, i
implies ha Neigh2𝑖
𝑙∩Neigh1𝑖
𝑙=∅.
The dis ance be ween nodes 𝑖 and 𝑥 in laye 𝑙 wi h 𝑥∈ 𝑁𝑒𝑖𝑔ℎ2𝑖
𝑙
alues 2.
𝑘𝑖𝑙
-
Numbe o neighbou s o node 𝑖 in laye 𝑙.
𝑘𝑖𝑙 equals he ca dinali y o Neigh1𝑖
𝑙 se .
103
AGoS𝑖𝑙
-
Con ibu ion o node 𝑖 o AGoS o laye 𝑙
𝐴𝐺𝑜𝑆𝑖𝑙=
{
1
𝑘𝑖𝑙∑1
1+𝑒−𝛽(Payo j
l−Payo 𝑖𝑙)
𝑗∈Neigh1𝑖
𝑙,𝑆𝑗𝑙=𝐶 ,𝑖𝑓 𝑆𝑖𝑙=𝐷
−1
𝑘𝑖𝑙∑1
1+𝑒−𝛽(Payo j
l−Payo 𝑖𝑙)
𝑗∈Neigh1𝑖
𝑙,𝑆𝑗𝑙=𝐷 ,𝑖𝑓 𝑆𝑖𝑙=𝐶
AGoS𝑙
-
AGoS in laye 𝑙
AGoS𝑙=1
𝑁𝑙∑AGoS𝑗𝑙
𝑁𝑙
𝑗=1
The AGoS a iables a e s o ed in an IEEE 754 double o ma wi h 8 by es, 52 bi s (plus 1
mo e implici bi ) o he signi ican digi s o a numbe in bina y o ma . This means a
p ecision o a leas 𝑓𝑙𝑜𝑜𝑟(𝑙𝑜𝑔10(253))=15 decimal digi s, mo e han enough o
accommoda e he cumula i e e ec s o ounding e o s esul ing om he i e a ions o each
simula ion.
In each i e a ion o he s ochas ic p ocess modelling he e olu ion o coope a ion on a
mul ilaye wi h 𝑀 laye s o ne wo ks, a e ocal node 𝑛 in laye 𝑙𝑛 has been selec ed, pai wise
compa ison o i s payo wi h he one o i s neighbou wi h a dis inc s a egy in he same laye
may dic a e wi h a Fe mi-like p obabili y ha s a egy 𝑆𝑛𝑙𝑛 is o be upda ed. In case i is and
be o e i is, he ollowing s eps a e aken in o de o inc emen ally upda e indi idual and global
payo s and AGoS wi h minimum CPU consump ion:
1. A e e y laye , dele e (i) con ibu ion o node 𝑛 o AGoS o he laye , (ii) con ibu ion o
di ec neighbou s o node n o AGoS o he laye and con ibu ions o nodes a dis ance
1 om node 𝑛 o (iii) he pa ial AGoS o nodes a dis ance 2 o node 𝑛 and (i ) o he
AGoS o he laye :
Fo e e y 𝑙∈{1,…,𝑀} do
AGoS𝑙←AGoS𝑙 ‐ AGoS𝑛𝑙 -- (i)
Fo e e y 𝑥∈Neigh1𝑛
𝑙 do
AGoS𝑙←AGoS𝑙 ‐ AGoS𝑥𝑙 -- (ii)
Fo e e y 𝑦∈Neigh2𝑛
𝑙 do
IF 𝑆𝑥𝑙≠𝑆𝑦𝑙 THEN
aux ← con ibu ion o 𝐴𝐺𝑂𝑆𝑦𝑙 om node 𝑥 in laye 𝑙
104
AGoS𝑦𝑙←AGoS𝑦𝑙−aux -- (iii)
AGoS𝑙←AGoS𝑙−aux -- (i )
END IF
2. Dele e con ibu ions o node 𝑛 o i s neighbou s’ payo s a (i) laye and (ii) mul ilaye
le el:
Se 𝐿𝑆={𝑙𝑛} o 𝐿𝑆={1,…,𝑀}, depending on in es men dis ibu ion c i e ia being
baseline o o he , espec i ely.
Fo e e y 𝑙∈𝐿𝑆 do
IF 𝑆𝑛𝑙==𝐶 THEN
Fo e e y 𝑥∈Neigh1𝑛
𝑙 do
aux ←𝐹2𝐼𝑛𝑣𝑒𝑠𝑡𝑛𝑙
Payo 𝑥𝑙←Payo 𝑥𝑙−aux -- (i)
Payo 𝑥←Payo 𝑥−aux -- (ii)
END IF
3. Upda e ocal node 𝑛 s a egy in laye 𝑙𝑛
𝑆𝑛𝑙𝑛←𝑆𝑛𝑙𝑛==𝐶 ?𝐷: 𝐶
4
Wi h s a egy 𝑆𝑛𝑙𝑛 upda ed, ollowing s eps close cu en i e a ion o s ochas ic p ocess:
4. In case o dis ibu ed in es men , i.e. no baseline, upda e in es men In es 𝑛
𝑙 ac oss
all 𝑙 laye s.
5. Upda e con ibu ions o node 𝑛 o i s neighbou s’ payo s a (i) laye and (ii) mul ilaye
le el:
Se 𝐿𝑆={𝑙𝑛} o 𝐿𝑆={1,…,𝑀}, depending on in es men dis ibu ion c i e ia being
baseline o o he , espec i ely.
4
𝑐𝑜𝑛𝑑 ? 𝑣𝑎𝑙1∶ 𝑣𝑎𝑙2 alues 𝑣𝑎𝑙1 o 𝑣𝑎𝑙2 depending on Boolean condi ion 𝑐𝑜𝑛𝑑 being
𝑡𝑟𝑢𝑒 o 𝑓𝑎𝑙𝑠𝑒, espec i ely.
105
Fo e e y 𝑙∈𝐿𝑆 do
IF 𝑆𝑛𝑙==𝐶 THEN
Fo e e y 𝑥∈Neigh1𝑛
𝑙 do
aux ←𝐹2𝐼𝑛𝑣𝑒𝑠𝑡𝑛𝑙
Payo 𝑥𝑙←Payo 𝑥𝑙+aux -- (i)
Payo 𝑥←Payo 𝑥+aux -- (ii)
6. Calcula e new node 𝑛 payo s (i) pe laye and (ii) mul ilaye :
Payo 𝑛←0
Fo e e y 𝑙∈{1,…,𝑀} do
Payo 𝑛
𝑙←0
Fo e e y 𝑥∈Neigh𝑛
𝑙 do
Payo 𝑛
𝑙← Payo 𝑛
𝑙+(𝑆𝑥𝑙==𝐶 ? 𝐹2In es 𝑥𝑙∶ 0)
Payo 𝑛
𝑙← Payo 𝑛
𝑙+(𝑆𝑛𝑙==𝐶 ?( 𝐹2‐1)In es 𝑛
𝑙𝑘𝑛𝑙∶ 0) -- (i)
Payo 𝑛←∑Payo 𝑛
𝑙
𝑀
𝑙=1 -- (ii)
7. Upda e (i) node 𝑛 AGoS in e e y laye and impac i on (ii) AGoS o he laye :
Fo e e y 𝑙∈{1,…,𝑀} do
Calcula e 𝐴𝐺𝑜𝑆𝑛𝑙 -- (i)
AGoS𝑙←AGoS𝑙 + AGoS𝑛𝑙 -- (ii)
8. A e e y laye , (i) calcula e AGoS o neighbou s o node 𝑛, (ii) impac i on he AGoS
o he laye , add con ibu ions o nodes a dis ance 1 om node 𝑛 (iii) o he pa ial
AGoS o nodes a dis ance 2 o node 𝑛 and (i ) o he AGoS o he laye :
Fo e e y 𝑙∈{1,…,𝑀} do
Fo e e y 𝑥∈Neigh1𝑛
𝑙 do
Calcula e 𝐴𝐺𝑜𝑆𝑥𝑙 -- (i)
AGoS𝑙←AGoS𝑙 + AGoS𝑥𝑙 -- (ii)
Fo e e y 𝑦∈Neigh2𝑛
𝑙 do
IF 𝑆𝑥𝑙≠𝑆𝑦𝑙 THEN
106
aux ← con ibu ion o 𝐴𝐺𝑂𝑆𝑦𝑙 om node 𝑥 in laye 𝑙
AGoS𝑦𝑙←AGoS𝑦𝑙+aux -- (iii)
AGoS𝑙←AGoS𝑙+aux -- (i )
END IF
Model a iables a e eady o subsequen s ochas ic p ocess i e a ion.

107
Appendix B. Deg ee-Deg ee Co ela ion and O e lapping
Deg ee-deg ee co ela ion in a mul ilaye is only meaning ul o i s mul iplex sub ype wi h
he same se o nodes p esen in all laye s, no in e -laye links, and in case hese a e suppo ed
on he e ogeneous ne wo ks.
Pea son co ela ion index is used o es ima e he deg ee co ela ion be ween wo laye s. I
a ies be ween -1 (when indi iduals ha e he same deg ees bu in di e en ne wo k laye s),
passing by 0 (meaning laye ne wo ks wi h unco ela ed indi idual deg ees) o 1 (when
indi iduals ha e coinciding deg ees in bo h laye ne wo ks). A me hodology o co ela ion
uning was de eloped, based on a simula ed annealing me hod p oposed in (Nicosia & La o a,
2015). The s a ing poin is a se o wo laye s, second one eplica ed om i s one. Then
i e a i ely, one andomly selec s a pai o indi iduals, 𝑁1 and 𝑁2, in second laye and swi ches
hei names in ha laye . Implici ly, in second laye 𝑁1‘s neighbou s swap wi h 𝑁2‘s and ice-
e sa. I he indi idual swi ching mo es deg ee-deg ee co ela ion owa ds in ended a ge i
is accep ed; o he wise, i is accep ed condi ioned on a ce ain p obabili y. This allowed
de ia ion om he pa h owa ds a ge co ela ion is essen ial, in o de o a oid ge ing s uck
on local minima and o keep he pa ame e space o exploi a ion open. This logic is applied in
cascade o he 𝑀−1 consecu i e pai s o an M-laye s mul ilaye .
As in (Nicosia & La o a, 2015), a M-laye s mul ilaye ha ing a pa icula deg ee-deg ee
co ela ion means ha laye s 𝑙 and 𝑙+1, wha e e 𝑙 𝜖 {1,…,𝑀−1} ha e ha deg ee-deg ee
co ela ion. As co ela ion be ween laye s is non- ansi i e, he pai s o laye s {𝑙−1,𝑙} and
{𝑙,𝑙+1} ha ing he same co ela ion does no imply he same alue applies o {𝑙−1,𝑙+1}
pai o laye s. Addi ionally and due o he inexis ence o in e -laye links, he no ion o
consecu i e laye s is pu ely a bi a y, depends on labelling, as laye s a e s a is ically
Figu e B-1- Example o a Deg ee-Deg ee Co ela ion Ma ix
108
indis inguishable om each o he . I deg ee-deg ee co ela ion 𝑀𝑥𝑀 ma ix is o be plo ed
o a M-laye mul ilaye esul ing om he applica ion o his algo i hm, main diagonal alues
1 and diagonals nex o main one assume mul ilaye deg ee-deg ee co ela ion. The ma ix is
symme ic. Values o o he en ies a e no an icipa ed. An example o a co ela ion 𝑀𝑥𝑀
ma ix is depic ed in Figu e B-1.
O e lapping is a di e en concep om deg ee-deg ee co ela ion, al hough no
independen . O e lapping be ween wo laye s 𝛼 and 𝛼′ p o ide an indica o on how p obable
i is o an a bi a y pai o nodes o be linked in bo h laye s. I is de ined as
𝑜𝑣𝑒𝑟𝑙𝑎𝑝𝑝𝑖𝑛𝑔𝛼𝛼′= ∑𝑎𝑖𝑗
𝛼𝑎𝑖𝑗
𝛼′
𝑖<𝑗
∑𝑎𝑖𝑗
𝛼+ ∑𝑎𝑖𝑗
𝛼′
𝑖<𝑗 − ∑𝑎𝑖𝑗
𝛼𝑎𝑖𝑗
𝛼′
𝑖<𝑗𝑖<𝑗
(1)
wi h
𝑎𝑖𝑗
𝛼= {1,𝑖𝑓 𝑛𝑜𝑑𝑒𝑠 𝑖 𝑎𝑛𝑑 𝑗 𝑎𝑟𝑒 𝑙𝑖𝑛𝑘𝑒𝑑 𝑖𝑛 𝑙𝑎𝑦𝑒𝑟 𝛼
0,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(2)
Al hough o e lap concep applies o mul ilaye s bo h wi h he e ogeneous ne wo ks as wi h
Ho and ne wo ks, o me case is he only one add essed because o i s inc eased ele ance
due o deg ee he e ogenei y. O e lapping be ween wo laye s was uned by copying andomly
links om one BA laye o he o he , un il eaching he p e ended numbe o eplica ed links.
Then, in he laye in cons uc ion new links a e added such ha (i) hey a e absen om
e e ence laye , (ii) each node needs o ha e a leas 𝑚 (=<𝑘>
2) connec ions and (iii)
p e e en ial a achmen p inciple is aken in o accoun when deciding on he nodes o be
connec ed nex by new links. The s eps ha e been conduc ed ensu ing ha ne wo ks emain
ully connec ed.
Deg ee-deg ee co ela ion and o e lapping a e no independen a iables, because one
canno be o ally con olled wi hou impac ing he o he , e.g. a pe ec 1 o e lapping implies
a deg ee-deg ee co ela ion o 1.
I laye s 𝛼 and 𝛼′ ha e no a single link in common connec ing he same pai o nodes,
o e lapping is null. I any pai o nodes linked in one laye is also linked in he o he ,
o e lapping is o al.
Along his appendix, he in luence o deg ee-deg ee co ela ion and o e lapping on
mul ilaye beha iou is aken in o accoun . He e we will conclude ha in gene al deg ee-
deg ee co ela ion e ains coope a ion and in pa icula o uncons ained (baseline) c i e ia
a ou s opology ensla emen , a phenomenon in which he sys em ge s insensi i e o sys em
109
pa ame e s as enhancemen ac o (𝐹) o in ensi y o selec ion (

) in he case o Public Goods
Games o alike, S o T in o iginal pai wise games as P isone ’s Dilemma. No ma e he alue
o his pa ame e , he le el o coope a ion achie ed coincides wi h he one he mul ilaye was
ini ialized wi h. O e lapping also leads o opology ensla emen because i implies deg ee-
deg ee co ela ion. Fo he cons ained case, opology ensla emen is also obse ed bu d i en
by a chain o mechanisms o di e en so desc ibed in appendix C.
In an a emp o shed ligh on deg ee-deg ee co ela ion in luence on coope a ion e olu ion,
a e age le el o coope a ion ac oss laye s was calcula ed as a unc ion o deg ee-deg ee
co ela ion and in ensi y o selec ion  o 8-laye s mul ilaye wi h BA ne wo ks and an
enhancemen ac o o 1.7, a alue o which coope a ion is iable o any in es men c i e ia.
The esul s disc imina ed by in es men dis ibu ion c i e ia a e p esen ed in igu e B-2.
Figu e B-2- A e age Le el o Coope a ion o Deg ee-Deg ee Co ela ion e sus In ensi y o Selec ion (𝛽).
Mul ilaye ha e 8 laye s, 1000 (𝑁) indi iduals pe laye , 〈k〉 equals o 4 and F alues 1.7 and BA ne wo ks. In
each laye , nodes we e andomly ini ialized wi h hal as Coope a o s.
The igu e highligh s wo ac s. The mos no o ious one is ha , apa om baseline c i e ia
o e y low in ensi y o selec ions, le el o coope a ion dec eases when deg ee-deg ee
co ela ion ge s s onge , a beha iou due o cumula i e payo . When deg ee-deg ee
co ela ion is maximum, he coope a ion le el o all c i e ia eaching he alue o 50%,
p ecisely he p opo ion o Coope a o s he mul ilaye was ini ialized wi h, is he second ac
o highligh in he igu e. This mul ilaye ine ia in changing ini ial le el o coope a ion had
al eady been no iced by (Kleinebe g & Helbing, 2018) o he baseline case.
The way a mul ilaye is cons uc ed, i ha ing a high deg ee-deg ee (Pea son) co ela ion
means ha he numbe o neighbou s any node has in any laye wi h a high p obabili y is
simila o he se o neighbou s he same node has ac oss all laye s in he mul ilaye . Thus,
odds o a hub in a laye being also a hub in ano he laye inc ease wi h deg ee-deg ee
110
co ela ion. Le us now conside a hub node and a high deg ee-deg ee co ela ion be ween he
laye s in he mul ilaye . Because o his co ela ion ac o , hubs a e aligned ac oss laye s. The
payo collec ed in laye s in which he node coope a es, because o i being cumula i e,
impac s all laye s. This means ha a hub can de ec in a laye 𝑙0 wi h a high payo collec ed
in o he laye s whe e he hub coope a es. The De ec o s a egy o his hub in laye 𝑙0 because
o i s high payo unc ions as a e e ence and ends o, i no o sa u a e he all laye as ALLD,
a leas o d y ou coope a ion in he neighbou hood. The ini ial s a egy o bigge nodes in a
laye , andomly de e mined, dic a es he di ec ion o sa u a ion, ALLC o ALLD, o he laye
o a leas he neighbou hood. Mo eo e , when deg ee-deg ee co ela ion is high, a node ends
o ha e he same neighbou s in all laye s, which means ha i i has k deg ee i ends o ha e
a o al o dis inc 𝑘 neighbou s ac oss all 𝑀 laye s o he mul ilaye .
Now le us conside a mul ilaye wi h maximum deg ee-deg ee co ela ion, in es men pe
laye c i e ia, an a bi a y ocal node and any one o i s neighbou s. These wo nodes will
sha e a link ac oss all laye s. I he neighbou is coope a i e in 𝑁𝐶 laye s, i will con ibu e o
ocal node payo wi h 𝐹
2𝑁𝐶 pe link in 𝑁𝐶links esul ing in an agg ega ed con ibu ion o 𝐹2. I
his neighbou now swi ches s a egy om coope a ion o de ec ion in one o he laye s i s
con ibu ion o ocal node payo will be upda ed o 𝐹
2(𝑁𝐶−1)(𝑁𝐶−1), i.e., in spi e o s a egy
upda e neighbou con ibu ion o ocal node payo was p ese ed, as long as he neighbou
kep coope a ing in a leas 1 laye , a condi ion wi h p obabili y ending o 1 as he numbe o
laye s ends o in ini y. This a ional is easily ex ended o ocal node change o s a egy and
he o he dis ibu ed in es men c i e ia. The bo om line is ha deg ee-deg ee co ela ion
ends o make indi iduals’ accumula ed payo s in a ian in ime. Addi ionally, ha ing
Coope a o s been andomly posi ioned, he chances o an indi idual A swi ching s a egy
unde in luence o a neighbou B in a laye is equal o chances o symme ical s a egy swi ch
be ween he same pai o indi iduals. This leads o he p ese a ion o ini ial le el o
coope a ion.
On he o he hand, o uncons ained baseline c i e ia, con ibu ions o a neighbou in
di e en laye s a e independen and an exclusi e unc ion o neighbou ’ s a egy in ha laye .
This sugges s baseline no o be so dependen on deg ee-deg ee co ela ion as igu e B-2
illus a es. Fo a ixed in ensi y o selec ion, he a ia ion o he le el o coope a ion e sus
deg ee-deg ee co ela ion eplica es indings in (Kleinebe g & Helbing, 2018). S ill o
baseline c i e ia and maximum deg ee-deg ee coope a ion, opological ensla emen de i es
om hub alignmen ac oss laye s, which p o ides hem a conside able s a egy ine ia,
117
I we conside equa ions 4 and 15, he co olla y ha ollows is ha in Ho and mul ilaye s
accumula ed payo a iance ends o ze o as he numbe o laye s inc ease. Expe imen al
e idences on i s momen s o accumula ed payo ac oss a Ho and mul ilaye a e depic ed
Figu e C-1- Payo s Dis ibu ion o Ho and Mul ilaye wi h In es men pe Game e sus Numbe o
Laye s. Ne wo ks a e buil wi h unco ela ed laye s wi h =1,〈𝑘〉=4,𝐹=1.7, 1000 nodes (𝑁) pe laye . .
Laye s we e ini ialized wi h hal nodes as Coope a o s andomly chosen. Resul s we e collec ed a e 100
gene a ions, one gene a ion being equal o numbe o laye s ies 1000 i e a ions.
Two ac s a e highligh ed in he igu e: payo a e age ends o 𝐹–1 and i s a ia ion o ze o as he numbe o
laye s inc ease. Al hough no p esen ed, he ex ension o his phenomenon o BA mul ilaye s was also no iced.
in igu e C-1.
Ha ing node payo a iance ending o ze o has an addi ional consequence when i comes
o a node o conside imi a ing a neighbou in a gi en laye wi h a di e en s a egy: he
a gumen o he exponen ial in Fe mi dis ibu ion ends o ze o, which allows i s exp ession
o be simpli ied o
𝑃𝑟𝑜𝑏(SA SB)=1
1+ 𝑒−𝛽(𝜋𝐵− 𝜋𝐴)≅12+𝛽4(𝜋𝐵−𝜋𝐴)
(16)
meaning ha wi h equal p obabili y he s a egy o a node is main ained o upda ed.
The p obabili y o a node upda ing i s s a egy on a laye depends now only on i and one
o i s neighbou s andomly chosen ha ing di e en s a egies, which depends on he le el o
coope a ion on he laye . Dependency on nodes ela i e payo anishes as he numbe o
laye s inc eases.
Up o now homogeneous ne wo ks wi h in es men pe game c i e ia we e conside ed, bu
a gumen s a e ex ensible o in es men pe laye c i e ia as a mul ilaye wi h homogeneous

118
ne wo ks wi h deg ee 〈𝑘〉, 𝛽0 in ensi y o selec ion and in es men pe game c i e ia beha es
exac ly he same as i i had in es men pe laye and 𝛽0/〈𝑘〉 in ensi y o selec ion.
As all nodes end o ha e a simila accumula ed payo and all laye s look and beha e alike,
i.e., hey a e eplicas om he same s ochas ic model, each gene ic laye can be mapped o a
one dimension andom walk alike model wi h dynamic p obabili ies o mo ing ei he way o
s aying in he same spo . Figu e C-2 plo s samples o ime se ies on he e olu ion o he
numbe o Coope a o s o bo h he a e age numbe o Coope a o s ac oss a Ho and
mul ilaye and he numbe o Coope a o s in indi idual laye s.
Figu e C-2- Time Se ies o E olu ion o Coope a ion Le el in 16 Laye Ho and Mul ilaye wi h In es men
dis ibu ed pe Game. Ne wo ks a e buil wi h =1,〈k〉=4,F=1.7, 1000 nodes (𝑁) pe laye . Laye s we e
ini ialized wi h hal nodes as Coope a o s. On he igh side mul ilaye a e age numbe o Coope a o s is plo
o 5 di e en uns. On he igh side, o a single un, he e olu ion o he numbe o Coope a o s pe laye is
plo . Sa u a ion o some laye s is no iceable.
Once a mul ilaye is cha ac e ized a a pa icula poin in ime, i s e olu ion depends
exclusi ely on i s s a us a ha poin in ime being i ele an he pa h leading o ha s a us.
This ac makes he e olu ion o he mul ilaye sui able o be s udied as a Ma ko p ocess in
which he pas has no in luence on he u u e once he p esen is speci ied. Thus, in a Ma ko
p ocess x( )
𝑃𝑟𝑜𝑏(𝑥(𝑡𝑛)

𝑥𝑛 | 𝑥(𝑡),𝑡

𝑡𝑛−1) = 𝑃𝑟𝑜𝑏(𝑥(𝑡𝑛)

𝑥𝑛 | 𝑥(𝑡𝑛−1))
(17)
𝑥(𝑡) is gene ically an a ay wi h one posi ion pe laye . A special so o Ma ko p ocess is
he Ma ko chain when he sys em can be desc ibed by a ini e o coun ably in ini e se o
s a es such ha he u u e e olu ion o he p ocess, once i is in a gi en s a e, depends only on
he p esen s a e and no on how i a i ed a ha s a e. A Ma ko chain is a s ochas ic model
ha can be desc ibed as a se o s a es, 𝑆 = {𝑠1,…,𝑠𝑛} and a se o e en s implying ansi ions
119
be ween s a es. The p ocess s a s in one o hese s a es and mo es successi ely om one s a e
o ano he wi h a p obabili y ha is an exclusi ely unc ion o he o me and la e s a es
i espec i ely o e en ual s a es isi ed be o e. Each mo e is called a s ep. I he chain is
cu en ly in s a e 𝑠𝑖, hen he p obabili y o mo ing o s a e 𝑠𝑗 is gi en by 𝑝𝑖𝑗. Na u ally, ∑𝑝𝑖𝑗𝑗
= 1. A single laye wi h N nodes can be modelled by a Ma ko chain whe e each s a e is
assigned a pa icula combina ion o s a egies ollowed by nodes. Fo a sys em wi h N nodes
and S s a egies he e a e po en ially 𝑁𝑆 di e en s a es. A mul ilaye wi h M laye s can also
be ep esen ed by a Ma ko chain, bu he numbe o s a es sky ocke s o 𝑁𝑆𝑀.
In o de o ackle his complexi y and wi h no loss o gene ali y, a single laye will be
add essed ins ead as ep esen a i e o he se o all mul ilaye laye s, as all laye s a e eplicas
om a single agen -le el dynamics e e ence. Mo eo e , a mean- ield app oxima ion will be
used in which he iden i y o he indi idual nodes ollowing a pa icula s a egy will no be
add essed, bu ins ead only he numbe o nodes ollowing each s a egy will be accoun ed.
Wi h mean ield app oxima ion, a scena io whe e he laye can be di ided by a on ie
such ha on each side o he line all nodes ha e he same s a egy in e ms o s a e
ep esen a ion canno be dis inguished om ano he one whe e nodes wi h di e en s a egies
a e all andomly mixed.
Wi h mean- ield ep esen a ion, because ep oduc ion is modelled ia an imi a ion p ocess,
a each s ep o he p ocess in he laye whe e imi a ion happens he numbe o Coope a o s is
al e ed by a mos 1 uni . This implies 𝑝𝑖𝑗 = 0 o |𝑖 –𝑗|>1.
A s a e i is called abso bing i 𝑝𝑖𝑗=𝛿𝑖𝑗, i.e., once s a e i is en e ed i is exi ed wi h p obabili y
ze o. ALLC and ALLD, i.e., s a e 1000 and s a e 0 will be abso bing s a es. Non-abso bing
s a es a e quali ied as ansien .
The g aphical ep esen a ion o he Ma ko chain de i ed om mean- ield e olu ion o a
single laye is as in igu e C-3. This Ma ko chain co esponds o he one desc ibing he
classical s ochas ic D unka d’s walk p ocess (o Gamble ’s Ruin) (G ins ead & Snell, 1997)
Figu e C-3- Ma ko Chain co esponding o D unka d’s Walk P ocess
120
wi h he ollowing di e ences: he p obabili y o p ese ing he s a e is no ze o (𝑝𝑖𝑖≠0) and
he p obabili ies o changing s a e depends on p esen s a e. The wo abso bing s a es map o
home and ba s a es in D unka d’s walk.
Co esponding Ma ko ansi ion ma ix as desc ibed in 2.1.3 o ini e popula ions, is
gi en by he ollowing exp essions:
1 , 𝑖=𝑗=0
1 , 𝑖=𝑗=𝑁
𝑁−𝑖
𝑁 ∗ 𝑖
𝑁−1 ∗ P obFe mi (Sj  Si) , 𝑗=𝑖+1 𝐴𝑁𝐷 0<𝑖<𝑁
𝑝𝑖𝑗= 𝑖𝑁 ∗ 𝑁−𝑖
𝑁−1 ∗ P obFe mi (Sj  Si) , 𝑗=𝑖−1 𝐴𝑁𝐷 0<𝑖<𝑁
1−p𝑖,𝑖−1−p𝑖,𝑖+1 , 0<𝑖<𝑁
0 , 𝑖=0 𝐴𝑁𝐷 𝑗>0
, 𝑖=𝑁 𝐴𝑁𝐷 𝑗<𝑁
, |𝑗−𝑖|>1
(18)
wi h 𝑃𝑟𝑜𝑏𝐹𝑒𝑟𝑚𝑖(𝑆𝑗

𝑆𝑖) ep esen ing he Fe mi p obabili y o node i wi h s a egy 𝑆𝑖 copying
𝑆𝑗 s a egy om node j. 𝑃𝑟𝑜𝑏𝐹𝑒𝑟𝑚𝑖(𝑆𝑗

𝑆𝑖) is a unc ion o bo h node i and node j payo s.
As he numbe o laye s inc eases, all nodes end o sha e he same payo and
𝑃𝑟𝑜𝑏𝐹𝑒𝑟𝑚𝑖(𝑆𝑗

𝑆𝑖) end o 12. Le us ocus on he con en o 𝑝𝑖,𝑖+1, he p obabili y o
inc easing he numbe o Coope a o s. Fi s ac o , 𝑁−𝑖
𝑁, e lec s he p obabili y o i s chosen
node being a De ec o . Second ac o , 𝑖
𝑁−1, accoun s o he p obabili y o , gi en ha a
De ec o has al eady been chosen, om N – 1 nodes o choose om, nex node o selec is 1
o i Coope a o s a ailable. Clea ly he e mean- ield app oach is ollowed, as he conc e e
unde lying ne wo k is no aken in o accoun . Now ha nodes selec ed a e sui able o a
s a egy imi a ion, all i is lacking is a a ou able p obabili y om Fe mi dis ibu ion.
Ma ko ansi ion ma ix mixes abso bing and ansien s a es. Fo an abso bing Ma ko
chain P and a e canonicaliza ion is pe o med as in 2.1.3, one ob ains he undamen al ma ix
N = ∑𝑄𝑘
+∞
𝑘=0 = (𝐼−𝑄)−1 o P. The en y 𝑛𝑖𝑗 o N gi es he expec ed numbe o imes ha
he p ocess eaches he ansien s a e 𝑠𝑗 i i is s a ed in he ansien s a e 𝑠𝑖. This implies
ha he expec ed numbe o s eps be o e he chain is abso bed, gi en ha he chain s a s in
121
s a e 𝑠𝑖, is gi en by 𝑡𝑖, he i- h elemen o column ec o , wi h 𝑡=𝑁𝑐, whe e c is a column
ec o all o whose en ies a e 1 (G ins ead & Snell, 1997).
The p obabili y ha Ma ko chain will e ol e o abso bing s a e 𝑠𝑗 s a ing om ansien
sa e 𝑠𝑖 is gi en by 𝑏𝑖𝑗 en y o ma ix B esul ing om 𝐵=𝑁𝑅.
In wha ollows, Ma ko chain is applied o a single mul ilaye a e age laye wi h 1000
nodes whose le el o coope a ion esul s om he a e age o Coope a o s ac oss all laye s.
Figu e C-4 plo s on i s le panel he heo e ical quasi-s a iona y dis ibu ion o he mean-
ield app oxima ion o single laye p e ending o ep esen a mul ilaye wi h in es men
dis ibu ed pe game and a numbe o laye s ending o in ini y. This quasi-s a iona y
dis ibu ion e lec s he p obabili y o each ansien s a e being isi ed un il he mul ilaye
sa u a es in any o he abso bing s a es. Conside ing all s a es equally p obable o mul ilaye
ini ializa ion and he e godici y o he Ma ko chain, wha is plo in le side panel is jus
∑𝑛𝑖𝑗𝑖
∑𝑛𝑖𝑗𝑖𝑗 , whe e he denomina o is jus a no maliza ion ac o o ans o m he numbe o isi s
in a s a e in o a p obabili y. This co esponds o equally weigh each line o he undamen al
N ma ix, i.e., each possible s a ing s a e. The equal p obabili y o all ansien s a es is he
highligh ing esul .
Figu e C-4- Theo e ical esul s o a Mean-Field app oxima ion o a Laye wi h Nodes wi h equal Payo .
Quasi-s a iona y dis ibu ion, numbe o s eps un il eaching an abso bing s a e and p obabili y o eaching
each abso bing s a e a e plo o a mean- ield app oxima ion o a gene ic laye p e ending o ep esen a
mul ilaye wi h in es men dis ibu ed pe game and numbe o laye s ending o in ini y
Vec o , wi h ields calcula ed as 𝑡𝑖=∑𝑛𝑖𝑗𝑗 is depic ed in cen al panel. As expec ed, he
numbe o s eps inc eases u he away om abso bing s a e ini ial s a e is loca ed. Symme y
o he line esul s om p oblem symme y, 𝑝𝑖,𝑗=𝑝𝑁−𝑖,𝑁−𝑗.
122
Finally he igh panel depic s no malized bij o inal abso bing s a e j equal o 0 o 1000,
i.e., he p obabili y o he sys em ending up in each o he wo abso bing s a e as a unc ion o
s a ing s a e. Lines in g aph a e complemen a y because in an abso bing Ma ko chain
𝑄𝑛

0 as he numbe o s eps (n) inc eases, hus he p obabili y ha he p ocess will be
abso bed is 1, and 0 and 1000 a e he only possible abso bing s a es.
This quasi-s a iona y dis ibu ion uni o mi y o a gene ic laye implies an AGoS ending
o ze o which o ces a mul ilaye o p ese e he p opo ion o Coope a o s wi h which i was
ini ialized. On he o he hand, as ime un olds he na u al end is o indi idual laye s o
sa u a e ei he as ALLC o ALLD. Thus, he ini ial p opo ion o Coope a o s is e lec ed in
he p opo ion o laye s sa u a ed as ALLC.
This laye pola iza ion is o no su p ise, because his o e all mul ilaye D unka d’s walk
alike p ocess, ha ing abso bing s a es o ALLC o ALLD, is doomed o con e ge o one o
hem. In his s ochas ic p ocess, he p obabili y o con e gence o ALLC inal s a e equals he
p opo ion o ini ial Coope a o s wi h which he sys em was ini ialized, which jus i ies why
he ini ial p opo ion o Coope a o s is p ese ed and why quasi-s a iona y dis ibu ion o
s a es is uni o m.
In o de o compa e heo e ical and expe imen ally he in luence o ini ial Coope a o
p obabili y in mul ilaye e olu ion, a mul ilaye wi h 16 laye s o BA and Ho and ne wo ks
wi h in es men c i e ia dis ibu ed pe game was independen ly ini ialized wi h a a iable
concen a ion o Coope a o s. Usual enhancemen ac o s we e applied. The e alua ion was
pe o med bo h un il and a sa u a ion ime. The esul s a e depic ed in Figu e C-5.
Focusing on BA mul ilaye , un il sa u a ion, inal le el o coope a ion depends on ini ial
one wi h almos no dependency on enhancemen ac o . Would he numbe o laye s inc ease
and his dependency would comple ely ade away. Addi ionally, o ex eme ini ial
p obabili ies, a sha p ansi ion in inal le el o coope a ion is no iceable. Conside ing again
a single laye wi h equal payo s ep esen a i e o he mul ilaye , heo e ical alue o inal
le el o coope a ion esul s om plo ing 𝐸(𝑁𝐶𝑓|𝑁𝐶𝑖), wi h 𝑁𝐶𝑖 and 𝑁𝐶𝑓 ep esen ing,
ecpec i ely, ini ial and inal numbe o Coope a o s. Taking in o accoun ha gene ic 𝑛𝑖𝑗
en y om Ma ko chain undamen al N ma ix ep esen s he expec ed numbe o imes he
mul ilaye will be in s a e j , gi en ha i s a s in s a e i, one has
P ob(𝑁𝐶𝑓=𝑗|𝑁𝐶𝑖=i) = 𝑛𝑖𝑗
∑𝑛𝑖𝑥
𝑥
(19)

123
𝐸(𝑁𝐶𝑓|𝑁𝐶𝑖=𝑖)=∑𝑗 𝑛𝑖𝑗
∑𝑛𝑖𝑥𝑥
𝑗
(20)
The heo e ical esul o 𝐸(𝑁𝐶𝑓|𝑁𝐶𝑖=𝑖) is p ecisely wha is plo on he op igh panel o
he igu e. The ma ching is pe ec pa icula ly o ex eme ini ial le els o coope a ion.
A sa u a ion ime he hea map is analogous apa om he ac ha inal le el o
coope a ion a ies linea ly wi h ini ial one. Agains his is as expec ed and al eady depic ed
in igh panel o igu e C-4. Wha happens o his BA mul ilaye as will be illus a ed in
appendix D when s udying i s AGoS is ha all laye s o he mul ilaye will sa u a e, some as
ALLC; o he s as ALLD. Mo eo e , because he naming o he laye s is a bi a y and laye s
a e independen , he same laye sa u a es ei he as ALLD o ALLC ac oss expe iences.
Figu e C-5- Topological and C i e ia Ensla emen in Mul ilaye s wi h 16 Laye s, In es men dis ibu ed pe
Game. Bo h BA and Ho and mul ilaye s ha e 1000 nodes (𝑁) and 〈k〉=4. Fo me one has =
0.05 and 〈k〉=4, la e one =1.0. Mul ilaye laye s a e independen and andomly ini ialized wi h a
numbe o Coope a o s aken om a disc e e andom a iable in he se 0 o 1000, inclusi e.
The p opo ion be ween he numbe o laye s in di e en condi ions will be such as dic a ed
by ini ial le el o coope a ion. Topology and in es men c i e ia ensla e he mul ilaye
because i s e olu ion canno be s ee ed by ac ing upon enhancemen ac o .
Reasoning on he basis o he Ma ko chain co esponding o a D unka d’s walk s ochas ic
p ocess, he single laye he mul ilaye is mapped o is doomed o end up as ei he ALLC
(𝑁𝐶𝑓=𝑁) o ALLD (𝑁𝐶𝑓=0), The p obabili y o ending up as ALLC is gi en by he ini ial
p opo ion o Coope a o s in he laye as om he igh panel om igu e C-4, i.e.,
124
𝑃𝑟𝑜𝑏(𝑁𝐶𝑓=𝑁)=𝑁𝐶𝑖
𝑁
(21)
As he e a e only wo ou comes possible, ALLC o ALLD,
E(𝑁𝐶𝑓)=𝑁𝑃𝑟𝑜𝑏(𝑁𝐶𝑓=𝑁)+0𝑃𝑟𝑜𝑏(𝑁𝐶𝑓=0)=𝑁𝐶𝑖
(22)
which leads one o conclude ha inal le el o coope a ion equals ini ial one.
As in ac one has a mul ilaye ins ead o a single laye and as all laye s sa u a e, he ini ial
p opo ion o Coope a o popula ion in he mul ilaye dic a es he pe cen age o laye s ha
sa u a e as ALLC, i.e., he inal p opo ion as Coope a o s in he sys em as all he o he laye s
will sa u a e as ALLD.
The Ho and mul ilaye exhibi s he same ends as desc ibed o BA mul ilaye bu wi h
de ia ions o small ini ial le els o coope a ion, pa icula ly o small alues o enhancemen
ac o s. This de ia ion om he heo e ical expec a ions is no su p ising and will be
expe ienced in ollowing igu es whe e i will be explained. We e he numbe o laye s g ea e
o he in ensi y o selec ion lowe and he ag eemen be ween heo e ical and expe imen al
esul s would be be e .
Conside ing exclusi ely mul ilaye s wi h in es men pe game c i e ia, he lines e lec ing
he way he numbe o laye s, in ensi y o selec ion, ype o ne wo k o enhancemen ac o
modula e he inal le el o coope a ion e sus ini ial a e depic ed in Figu e C-6. The highe
Figu e C-6- Final Le el o Coope a ion as a Func ion o ini ial One. Each ne wo k has 1000 nodes (𝑁), 〈𝑘〉=
4. In es men is dis ibu ed pe game and laye s in he mul ilaye we e ini ialized wi h hal nodes as
Coope a o s.
125
he numbe o laye s o a ixed in ensi y o selec ion, he close expe imen al cu e i s
heo e ical one. Highe alues o in ensi y o selec ion equi e high numbe o le els o
expe imen al lines o be e i heo e ical ones. This makes sense because in he exponen ial
a gumen con olled o he Fe mi dis ibu ion con olling he p obabili y o s a egy imi a ion
an inc ease in he β alue is compensa ed by a lowe payo a iance esul ing om a highe
numbe o laye s and ice- e sa. Highe alues o enhancemen ac o s a e pa icula ly use ul
o low ini ial le el o coope a ion. Some hing also expec able as i is essen ial o he ew
Coope a o s o hold hei g ound.
Theo e ical and expe imen al lines sha e he same shape bu he de ia ion be ween hem is
highe o smalle ini ial le els o coope a ion. This is no su p ising because he heo e ical
model is based on he app oxima ion o a binomial dis ibu ion o a Poisson one, which is only
alid o g ea alues o

(=𝑁𝑝). Fo smalle alues o ini ial le el o coope a ion,
heo e ical model loses alidi y and so i is senseless o expec an exac ma ch be ween hese
lines.
Taking as a e e ence in es men dis ibu ion pe game c i e ia, o a gi en ype o ne wo k,
Ba a o Ho and, he numbe o laye s de e mines de a iance o accumula ed payo be ween
nodes, which dic a e he le el o opological and c i e ia ensla emen ..
Fo in es men dis ibu ed pe game, Figu e C-7 depic s he ac ha o a gi en numbe o
laye s, 16 in his pa icula case, a maximum in ensi y o selec ion can be iden i ied below
which mul ilaye e olu ion is s uck o ini ial condi ions and insensi i e o a ia ions in
enhancemen ac o . Fo Ho and ne wo ks, he igu e shows ha up o in ensi y ac o s aluing
1 and i espec i e o enhancemen ac o , a e age le el o coope a ion does no change om
ini ial alues. We e he mul ilaye ini ialized wi h a di e en alue and ha same alue would
Figu e C-7- Topological and C i e ia Ensla emen o 16-laye s Mul ilaye , 1000 nodes (𝑁) pe laye .
Mul ilaye s a e ini ialized wi h hal Coope a o s pe laye andom and independen ly selec ed.
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be p ese ed. Mo eo e , i hea maps co esponding o a di e en numbe o laye s a e
depic ed, i will be no iceable ha he wa e on o he colou code co esponding o he
ini ial le el o coope a ion wi h which he mul ilaye was ini ialized mo es igh wa ds as he
numbe o laye s inc ease.
Fo a dis ibu ed in es men c i e ia, he e ogeneous ne wo ks end o ha e a beha iou
simila o homogeneous one, bu equi ing mo e laye s o achie e he same beha iou wi h
he same in ensi y o selec ion. Fo a gi en ne wo k ype, in es men pe laye c i e ia ends
o expe ience he same beha iou as i s in es men pe laye s coun e pa , bu wi h a highe
numbe o laye s and/o lowe in ensi y o selec ion.
De ining ensla emen as he condi ion o inal le el o coope a ion in a mul ilaye wi h
in es men pe game c i e ia di e ing in less han 20% om he ini ial le el o 50% wi h
which a mul ilaye was ini ialized, wha e e he alue o enhancemen ac o 𝐹∈[1,2], Figu e
C-8 displays he domain o he numbe o laye s e sus in ensi y o selec ion 𝛽 whe e
ensla emen ules. We e he ensla emen c i e ia mo e demanding agains accep able
luc ua ion on he inal le el o coope a ion, o he same numbe o laye s a lowe in ensi y
o selec ion would be equi ed.
Figu e C-8 - Domain o Topological Ensla emen o In es men dis ibu ed pe Game. Shaded a eas
ep esen he locus o numbe o laye s l and in ensi y o selec ion 𝛽 pa ame e s such ha hea map ℎ𝑚𝑙(𝛽,𝐹)
displaying he le el o coope a ion achie ed in a mul ilaye wi h l laye s, ini ialized wi h hal nodes as
Coope a o s, in es men dis ibu ed pe game, in ensi y o selec ion 𝛽 and enhancemen ac o F, o a gi en
numbe o laye 𝛽=max
𝑥|ℎ𝑚𝑙(𝑥,𝐹)−0.5|<0.1,x∈[10−2,10],∀ 𝐹∈[1,2]
Node consis ency is ano he possible compa ison pe spec i e o con on a ion be ween
expe imen al and heo e ical esul s as depic ed in igu e C-9. Due o he ini e numbe o
laye s i mus be s essed ha expe imen al consis ency e ol es by quan a ha amoun o
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Appendix E. Mul ilaye s wi h di e en Types o Ne wo ks
In he main body o he documen , mul ilaye s we e conside ed o independen laye s
suppo ed on ne wo k ins ances o a single ype. He e we d op second cons ain: laye s a e
s ill independen bu need o mo e o sha e ne wo k ype. Figu e E-1 p esen s esul s collec ed
om a mul ilaye wi h hal laye s as Ho and, ano he hal as BA.
Figu e E-1- Mul ilaye wi h 8 laye s BA o Ho and s. Mixed Mul ilaye wi h 4 laye s Ho and plus 4 laye s
BA. Le el o coope a ion and AGoS o pu e 8-laye Ho and and BA mul ilaye a e compa ed wi h a mixed
mul ilaye wi h hal laye s wi h Ho and ne wo ks and he o he hal wi h BA ne wo ks. AGoS lines we e
calcula ed o in ensi y o selec ion 𝛽=1, enhancemen ac o 𝐹=1.6 and 1000 (𝑁) nodes pe laye .
Laye s we e independen ly ini ialized wi h hal indi iduals as Coope a o s o a numbe o Coope a o s gi en by
a andom a iable uni o mly dis ibu ed be ween 0 and 1000, inclusi e, depending on he le el o coope a ion
o AGoS being calcula ed, espec i ely.
Lines o single ne wo k ype mul ilaye s we e eco e ed om g aphics p e iously aced
on documen main body. Pu e BA mul ilaye s a e mo e coope a i e han Ho and ones. In
pa icula o enhancemen ac o 𝐹=1.6 and baseline c i e ia, Ho and only 8-laye
mul ilaye is de ec i e. Howe e , in he mixed scena io and due o con ibu ions o
accumula ed payo ecei ed om o some BA laye s coope a ion becomes easible. As a
gene al ule, he beha iou o he mixed mul ilaye is posi ioned somewhe e be ween
beha iou s o pu e ones.

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