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Unique supe di usion induced by di ec ionali y in mul iplex ne wo ks
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PAPER
Unique supe di usion induced by di ec ionali y in mul iplex
ne wo ks
Xiang ong Wang1,2, Alejand o Tejedo 3,4,YiWang
1,2and Yami Mo eno5,6,7,∗
1Ins i u e o Fu u e Ne wo ks, Sou he n Uni e si y o Science and Technology, Shenzhen, People’s Republic o China
2Resea ch Cen e o Ne wo ks and Communica ions, Peng Cheng Labo a o y, Shenzhen, People’s Republic o China
3Depa men o Science and Enginee ing, So bonne Uni e si y Abu Dhabi, Abu Dhabi, Uni ed A ab Emi a es
4Depa men o Ci il and En i onmen al Enginee ing, Uni e si y o Cali o nia I ine, I ine, CA, Uni ed S a es o Ame ica
5Ins i u e o Biocompu a ion and Physics o Complex Sys ems, Uni e si y o Za agoza, Za agoza, Spain
6Depa men o Theo e ical Physics, Uni e si y o Za agoza, Za agoza, Spain
7ISI Founda ion, Tu in, I aly
∗Au ho o whom any co espondence should be add essed.
E-mail: yami .mo en[email p o ec ed]om
Keywo ds: ne wo ks, mul iplex, di ec ionali y, complex sys ems, spec al p ope ies
Abs ac
The mul ilaye ne wo k amewo k has se ed o desc ibe and unco e a numbe o no el and
un o eseen physical beha io s and egimes in in e ac ing complex sys ems. Howe e , he majo i y
o exis ing s udies a e buil on undi ec ed mul ilaye ne wo ks while mos complex sys ems in
na u e exhibi di ec ed in e ac ions. He e, we p opose a amewo k o analyze di usi e dynamics
on mul ilaye ne wo ks consis ing o a leas one di ec ed laye . We igo ously demons a e ha
di ec ionali y in mul ilaye ne wo ks can undamen ally change he beha io o di usi e
dynamics: om mono onic (in undi ec ed sys ems) o non-mono onic di usion wi h espec o
he in e laye coupling s eng h. Mo eo e , o ce ain mul ilaye ne wo k configu a ions, he
di ec ionali y can induce a unique supe di usion egime o in e media e alues o he in e laye
coupling, whe ein he di usion is e en as e han ha co esponding o he heo e ical limi o
undi ec ed sys ems, i.e. he di usion in he in eg a ed ne wo k ob ained om he agg ega ion o
each laye . We heo e ically and nume ically show ha he exis ence o supe di usion is ully
de e mined by he di ec ionali y o each laye and he opological o e lap be ween laye s. We
u he p o ide a o mula ion o mul ilaye ne wo ks displaying supe di usion. Ou esul s
highligh he significance o inco po a ing he in e ac ing di ec ionali y in mul ile el ne wo ked
sys ems and p o ide a amewo k o analyze dynamical p ocesses on in e connec ed complex
sys ems wi h di ec ionali y.
1. In oduc ion
Mul ilaye ne wo ks p o ide a p ope ma hema ical ep esen a ion o complex sys ems wi h di e en ypes
o in e ac ions, e.g. social in e ac ions among indi iduals ac oss di e en social pla o ms such as Facebook
and Twi e , and enable he unde s anding o dynamics ac ing on o e ol ing wi hin hose sys ems [1,2]. A
numbe o dynamical p ocesses a e s udied on he amewo k o mul ilaye ne wo ks: o example,
inno a ion/in o ma ion di usion [3], he con ol o o ma ion and ask alloca ion o swa ms [4–6],
synch oniza ion o oscilla o s [7], he unde s anding o unc ional b ain connec i i y [8], and many o he s.
Mo e specifically, ecen s udies based on a mul ilaye ne wo k ep esen a ion ha e e ealed un o eseen
dynamical egimes and beha io s, such as new sp eading egimes when mo e han one diseases in sp eading
in a gi en popula ion [9–12], and enhanced s abili y o synch onized s a es [13]. Mo eo e , con e sely om
building up a mul ilaye s uc u e, g adually disman ling a mul ilaye ne wo k igge s an ab up ansi ion
[14], sugges ing a non-addi i e e ec o a mul ilaye s uc u e om he in eg a ion o i s laye s. In e ms o
di usion dynamics, mul ilaye ne wo k s udies e ealed ha o any mul ilaye configu a ion, he di usi e
beha io o he o e all sys em (mul ilaye ne wo k) could be uned o be as e han ha o he slowes laye
© 2021 The Au ho (s). Published by IOP Publishing L d on behal o he Ins i u e o Physics and Deu sche Physikalische Gesellscha
New J. Phys. 23 (2021) 013016 XWange al
o significan alues o in e laye di usi i y (in compa ison o in alaye di usi i y). Mo e su p isingly, bu
only o ce ain mul ilaye ne wo k configu a ions, he o e all sys em can exhibi a supe di usi e beha io
o la ge alues o in e laye di usi i y, whe e he di usi e dynamics on he mul ilaye ne wo k is as e
han in any o he indi idual laye s, when hose a e conside ed in independen ly [15–17].
Mos o he s udies made in he field o mul ilaye ne wo ks adop undi ec ed s uc u es, i.e. he
in e ac ions ep esen ed by bo h in e laye and in alaye links a e assumed o be undi ec ed. Howe e ,
mos eal wo ld sys ems a e inhe en ly di ec ed. Examples a e he Wo ld Wide Web, in which hype links
uninonedi ec ion omoneWebpage oano he ; oodwebs,inwhichene gyflows omp ey o
p eda o ; phone call ne wo ks; me abolic ne wo ks; ci a ion ne wo ks; social ne wo ks o ollowe s and
ollowee; gene egula ion ne wo ks, o name a ew. Addi ionally, some dynamical p ocesses a e go e ned by
physical laws and inhe en ly e ol e in a di ec ional manne despi e o he unde lying undi ec ed
connec i i y opology, like cu en /in o ma ion flow om high ol age o low ol age, i us ansmissions
om in ec ed o suscep ible indi iduals. Mo eo e , mul i-le eled unidi ec ional in e ac ions a e commonly
ound in in e ac ing sys ems. Fo example, a gene egula ion ne wo k aims o p ope ly cha ac e ize bo h
p o ein-DNA and p o ein-p o ein di ec ed in e ac ions be ween genes and p o eins [18]. The ein,
mul ilaye ne wo ks wi h di ec ionali y p o ide a ealis ic ep esen a ion o such sys ems consis ing o
mul ile el and di ec ed in e ac ions.
Mul ilaye ne wo ks whe e a leas one o he laye s is di ec ed gi e ise o new heo e ical challenges in
analyzing hei dynamical beha io s. As di usion dynamics on undi ec ed mul ilaye ne wo ks is ully
cha ac e ized by he spec um o ne wo k ma ices, like sup a-Laplacian [15], ne wo k di ec ionali y leads
o nonsymme ical g aph ma ices, which mos ly ea u es complex and non-o hogonal spec um.
Consequen ly, spec um ela ed analysis on undi ec ed mul ilaye ne wo ks migh be p o en as in alid o
inaccu a e. Few ecen s udies on di ec ed mul ilaye ne wo ks e eal new and unexpec ed physical
beha io s [16,19], as well as inc eased sensi i i y o s uc u al changes [20].
The con ibu ion o his pape is o p o ide a heo e ical amewo k o di usi e dynamics on
mul ilaye ne wo ks wi h di ec ed laye s. We show ha al hough ce ain undi ec ed mul ilaye ne wo ks can
esul in enhanced di usi e ( as e han he slowes laye ) o supe di usi e ( as e han any o he laye s)
egimes o la ge alues o he in e laye coupling, he di usi e a e o he o e all sys em ne e exceeds he
di usion on he equi alen agg ega ed sys em ( he di ec sum o each laye ). We also show ha ce ain
di ec ed mul ilaye ne wo ks can also show enhanced di usi e and supe di usi e egimes o la ge alues o
in e laye coupling. Rema kably, di ec ed mul ilaye ne wo ks, di e en ly om hei undi ec ed
coun e pa s, can exhibi a unique supe di usi e beha io o in e media e alues o coupling, whe e
di usion dynamics a e e en as e han he co esponding agg ega ed sys em. This unique supe di usion
egime only eme ges when ce ain s uc u al condi ions a e sa isfied. We analy ically and nume ically
unco e condi ions d i ing he unique supe di usion in di ec ed mul ilaye ne wo ks. In addi ion, we
p opose a model o configu e di ec ed mul ilaye ne wo ks ha achie es he unique supe di usion wi h
unable magni ude.
The pape is o ganized as ollows. Di usi e dynamics on mul ilaye ne wo ks wi h di ec ion is desc ibed
in he Model sec ion 2.Thesec ion3o Resul s p esen s he condi ions o di usion egimes below he
di usi i y in he co esponding agg ega ed sys em in sec ion 3.1 and o he supe di usion egime abo e
he di usi i y in he agg ega ed sys em in sec ion 3.2, ollowed by a model o he cons uc ion o a
mul ilaye ne wo k wi h supe di usion in sec ion 3.3.Sec ion4concludes he wo k.
2. Model
Mul ilaye ne wo ks consis o nodes in e ac ing bo h wi hin he same laye and ac oss di e en laye s ia
in alinks and in e links which encodes mul iple ypes o in e ac ions. In his s udy, we ocus on a pa icula
ype o mul ilaye ne wo ks called mul iplex ne wo ks, cha ac e ized by laye s consis ing o he same se o
nodes, bu possibly di e en connec i i y (laye opology); and laye s in e ac ing wi h each o he only ia
coun e pa node. The opological s uc u e o he mul iplex is desc ibed by he so-called sup a-adjacency
ma ix A=aij, which is a block diagonal ma ix. Each block is an N×Nma ix, whe e Nis he numbe
o nodes pe laye . Each diagonal block co esponds o he adjacency ma ix o a laye (in a-laye
connec i i y, and he e o e an en y aij =1 in such a block co esponds o an in e laye link), while he
o -diagonal blocks, gi en he defini ion o a mul iplex ne wo k, a e jus N×Niden i y ma ices,
ep esen ing he in e -laye links be ween eplica nodes ac oss laye s. No e ha a di ec ed mul iplex, is a
mul iplex whe e a leas one o he diagonal blocks is no symme ic.
Le Qbe he co esponding Laplacian ma ix, defined as Q=D−Awhe e D=diag (di)and
di=N
i=1aij deno ing he ou -going deg ee o a node i. We conside wi hou loss o gene ali y a mul iplex
ne wo k consis ing o wo di ec ed laye s and in e connec ed ia links o weigh p⩾0, he co esponding
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New J. Phys. 23 (2021) 013016 XWange al
Laplacian ma ix Qcan be w i en in a block o m as
Q=Q1+pI −pI
−pI Q2+pI(1)
whe e Qi=Di−Ai,i=1, 2, deno es he Laplacian ma ix o he di ec ed g aph in each laye and pI
encodes he weigh ed in e connec ions ac oss laye s. The abo e defined Laplacian ma ix o mul ilaye
ne wo ks wi h di ec ed laye s is non-symme ical, which migh lead o complex a he han eal spec um.
As he Laplacian sa isfies a ze o ow sum, i.e. Qu =0wi hudeno ing he all one ec o , alue 0 is an
eigen alue wi h igh eigen ec o uand le eigen ec o deno ed as y. Fo a nonsymme ic ma ix, le and
igh eigen ec o s a e in gene al no he same.
To analyze he e ec o ne wo k di ec ionali y on dynamical p ocesses, we conside he s anda d
di usi e dynamics on mul ilaye ne wo ks wi h di ec ed laye s and de e mine he con e gence a e o a
fixed poin solu ion. Le xi( ) deno es he s a e o a node ia ime . Changes o he s a e xi( )wi h espec o
ime is go e ned by ˙
xi( )=−dixi( )+iaijxi( ). The di usi e dynamics on a mul ilaye ne wo k can be
w i eninama ix o mas dx( )T
d =−x( )TQ(2)
whe e ()Tdeno es ansposi ion, and x( ) is he ec o encoding he s a e o all nodes a ime . The solu ion
o he abo e go e ning equa ion ollows x( )T=x(0)Te−Q .Thefixedpoin x∗Ta which ˙
x∗T=0is
calcula ed employing he conse a ion law x(0)Tu=x(∞)Tu,whe eallone ec o uis he igh eigen ec o
co esponding o eigen alue 0 o he Laplacian ma ix Q.Le ydeno e he le eigen ec o o Q
co esponding o eigen alue 0. Combining he conse a ion law x(0)Tu=x∗Tuand yTQ=0yields he
solu ion o fixed poin
x∗=x(0)Tu
yTuy.(3)
We show in appendix Aa con e gence o he fixed poin in equa ion (3) is gua an eed o a connec ed
di ec ed g aph. The con e gence a e o he fixed poin x∗is cha ac e ized by he second smalles eal pa
[16], deno ed as Re λ2(Q), o he eigenspec um o he Laplacian ma ix.
2.1. Gene al analysis o di usi e beha io
To analyze he con e gence beha io o di usi e dynamics on mul iplex ne wo ks wi h di ec ed laye s, we
s udy he spec um o he go e ning Laplacian ma ix and compa e wi h he in eg a ed sys em. The
cha ac e is ic polynomial o he Laplacian ma ix Qin equa ion (1) can be calcula ed by, applying he
Schu complemen heo em on he block Laplacian ma ix,
de (λI−Q)=de λI−Q1+Q2
2(λ−2p)I−Q1+Q2
2−ΔQ(4)
whe e 4ΔQ=(Q1−Q2)2−2(Q1Q2−Q2Q1). The eigen-pai o eigen alue 2pand eigen ec o (u,−u)
holds o di ec ed mul ilaye ne wo ks, which is p e iously ound in undi ec ed mul ilaye ed ne wo ks
[12,21]. The join e ec o he Laplacian ma ices Q1and Q2 o each laye , encoded in he ma ix ΔQ,
de e mines he de ia ion o he con e gence a e o he whole sys em om ha o he in eg a ed mul ilaye .
Depending on ela ions be ween he di usi e a e o he sys em as a whole and ha o he in eg a ed
sys em, we e e wo complemen a y egimes, supe di usi e and non-supe di usi e, as ollows: (i) i
Re λ2(Q) o he o e all sys em is g ea e han λ2((Q1+Q2)/2) o he in eg a ed sys em, we e e o as
supe di usi e, (ii) o he wise as non-supe di usi e.
Re λ2(Q)⎧
⎨
⎩
>λ
2((Q1+Q2)/2) Supe di usi e
⩽λ2((Q1+Q2)/2) Non-supe di usi e (5)
We men ion ha he supe di usi e egime defined in his wo k sligh ly di e s om he defini ion in
li e a u e [15,16], whe e supe di usion implies a as e di usion on he o e all sys em han he di usion
on he as es laye . He e, we mainly ocus on di usi e beha io compa ing he o e all sys em and he
co esponden in eg a ed sys em, gi en he la e se es as he asymp o ic di usion as in e coupling s eng h
goes o infini y and, mos impo an ly, se es as a heo e ical uppe limi o di usion on undi ec ed
mul iplex ne wo ks [as shown in he ollowing equa ion (8)].
Unde he special case o Q1=Q2, sol ing equa ion (4)yieldsReλ2(Q)=min λ2((Q1+Q2)/2), 2p
and he e o e only he non-supe di usi e egime is exhibi ed. Gene al cases o Q1=Q2 esul in
possibili ies o bo h non-supe di usi e and supe di usi e egimes. Figu e 1exemplifies h ee ypologies
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New J. Phys. 23 (2021) 013016 XWange al
Figu e 1. Supe di usi e and non-supe di usi e egimes on mul iplex ne wo ks wi h a leas one di ec ed laye s. Panel (a) shows
a mul iplex ne wo k consis ing o a di ec ed laye and an undi ec ed laye displaying a non-supe di usi e egime
(Re λ2(Q)<λ
2Q1+Q2
2). Panel (b) shows a mul iplex ne wo k wi h coupled laye s in a e e sed di ec ion, exhibi ing a
supe di usion (Re λ2(Q)>λ
2Q1+Q2
2) a e he ansi ion om a linea g ow h o 2p. Panel (c) shows a mul iplex wi h
coupled laye s in he same di ec ion, exhibi ing a non-supe di usi e egime (Re λ2(Q)=λ2Q1+Q2
2).
including (a) coupled di ec ed and undi ec ed laye s, (b) coupled di ec ed laye s wi h he opposi e
di ec ion, and (c) coupled di ec ed laye s wi h he same di ec ion. Figu e 1(a) and (c) show only
non-supe di usi e egimes and figu e 1(b) exhibi s supe di usion. Figu es 2(a) and (c) show he
non-supe di usi e and supe di usi e beha io on eal-wo ld mul iplex ne wo ks, espec i ely.
Gi en he ich dynamical beha io o di usion in di ec ed mul iplex, i is o pa amoun impo ance o
iden i y he key s uc u al p ope ies unde pinning each o he obse ed egimes. In his s udy, we mainly
ocus on iden i ying he unde lying condi ions d i ing he non-supe di usi e and supe di usi e egimes
employing he p incipal subma ix app oach and he Cauchy in e lacing heo em [22] (o he Poinca e
sepa a ion heo em). In addi ion, we employ he eigen alue p ope y o a no mal ma ix, which sa isfies
XXT=XTX. Fo a no mal ma ix, he eal pa s o eigen alues a e he eigen alues o he He mi ian pa
Re X=(X+XT)/2o hema ixX.
The Laplacian ma ix is simila o he ollowing ma ix 
Qin a ans o med basis

Q=1
√2II
−II
QI−I
II
1
√2(6)
which can be simplified as

Q=⎡
⎢
⎣
Q1+Q2
2
Q2−Q1
2
Q2−Q1
2
Q1+Q2
2+2pI
⎤
⎥
⎦.(7)
The in eg a ed Laplacian Q1+Q2
2appea s as a p incipal subma ix o he ans o med Laplacian 
Qo he
whole sys em. In addi ion, eigen alues o he ma ix 
Qa e he same wi h eigen alues o Qdue o he
simila i y ela ion be ween Qand 
Q. Applying he Cauchy in e lacing heo em on he ans o med ma ix

Qand he in eg a ed sys em Q1+Q2
2as a p incipal subma ix, undi ec ed mul ilaye ne wo ks always sa is y
he ollowing in e lacing ela ion
λ2(Q)⩽λ2Q1+Q2
2.(8)
Hence, he e ne e occu s a supe di usion in undi ec ed mul ilaye ne wo ks, ega dless o he s eng h o
in e coupling. Equa ion (8) se es as a heo e ical limi and implies ha al hough mul ilaye s uc u e o
undi ec ed laye s enhances he less di usi e laye , he di usi e a e is uppe bounded by he in eg a ed
sys em.
4

New J. Phys. 23 (2021) 013016 XWange al
Figu e 2. Non-supe di usi e and supe di usi e eal-wo ld mul iplex ne wo ks. Panel (a) shows a non-supe di usi e egime on
he Vicke s–Chan mul iplex social ne wo ks [31] wi h di ec ed laye s. Panel (c) shows a supe di usion on he mul ilaye ne wo k
whe e one laye is om Vicke s–Chan mul iplex and he o he laye is he e e se o he fi s laye . Panels (b) and (d) show he
dis ibu ion o all eigen alues o he Laplacian Q o he o e all mul iplex and ha o he in eg a ed sys em (Q1+Q2)/2
co esponding o ne wo ks in panels (a) and (c), espec i ely.
3. Resul s
3.1. Condi ions o he non-supe di usi e egime
Fo mul ilaye ne wo ks wi h di ec ed laye s, he condi ions o non-supe di usi e and supe di usi e
egimes a e ansla ed in o condi ions when he Cauchy in e lacing heo em be ween a non-symme ic
ma ix and i s p inciple subma ix is sa isfied. S a ing om he special case o coupled iden ical laye s, i
ollows
Re λ2(Q)=λ2Q1+Q2
2(9)
which implies a non-supe di usi e egime o di ec ed mul ilaye ne wo ks wi h iden ical di ec ed laye s o
p⩾λ2(Q1+Q2)/2/2. When he ans o med Laplacian 
Qand all he p incipal ma ices including he
in eg a ed ma ix (Q2+Q1)/2 a e no mal, he in e lacing is also sa isfied. Because o a p incipally no mal
ma ix, all eigen alues o ma ices Qand (Q2+Q1)/2 lie on he same complex line wi h an angle θin he
complex plane [23,24]. Fo a p incipally no mal ma ix, he pa ial o de o gene alized in e lacing on a
complex plain is ob ained:
|λ2(Q)|⩽λ2Q1+Q2
2.(10)
As he eal pa o eigen alue eads |λ2(Q)|eiθand θis he same o all eigen alues, i implies an in e lacing
be ween eal pa s o eigen alues and acco dingly a non-supe di usi e egime o mul ilaye ne wo ks wi h
p incipally no mal Laplacian ma ices.
When he equi emen o no mali y o all p incipal subma ices is elaxed o he no mali y o a single
p incipal subma ix, gene alized in e lacing o lexicog aphic o de can be applied o 
Qand (Q2+Q1)/2.
Fo a no mal Laplacian ma ix and a no mal p incipal subma ix Q1+Q2
2,weha e ha
λ2(Q)⩽θλ2Q1+Q2
2(11)
whe e ⩽θ ep esen s he lexicog aphic o de [25] cha ac e ized by he posi i e cone H:={a+bi :
a>0o a=0andb>0}.Equa ion(11)means ha e
−iθλ2Q1+Q2
2−e−iθλ2(Q)lies in a posi i e cone,
5
New J. Phys. 23 (2021) 013016 XWange al
which implies λ2Q1+Q2
2⩾Re λ2(Q). The e o e, di ec ed mul ilaye ne wo ks wi h no mal Laplacian
ma ix and no mal p incipal subma ix Q1+Q2
2exhibi only he non-supe di usi e egime.
3.2. Condi ions o he supe di usi e egime
Di usion on undi ec ed mul ilaye ne e exceeds he di usion o in eg a ed pa . Howe e , ce ain
mul ilaye ne wo ks wi h di ec ed laye s b eak he cons ain and exhibi a unique supe di usion. In his
subsec ion, we analyze condi ions o he occu ence o such supe di usion. Fo he gene alized in e lacing
heo em o hold, i equi es bo h he whole sys em and he in eg a ed sys em o be no mal o
simul aneously close o no mal. When he condi ion is elaxed such ha he in eg a ed sys em is no mal o
close o no mal, while he whole sys em de ia es om no mali y, i migh occu wi h supe di usion. We
p o ide analy ical e idences by fi s ly ela ing he eal pa o eigen alues and eigen alues o he He mi ian
pa o he o iginal ma ix, which ollows
N

j=N−k
Re λj=
N

j=N−k
x∗
jRe 
Qxj⩽
N

j=N−k
λjRe 
Q(12)
whe e Re 
Qdeno es he He mi ian pa o ma ix 
Q.BecauseN
j=1Re λj=N
j=1λjRe 
Qand
λ1Re 
Q=0 o a no mal subg aph, we ha e ha Re λ2(Q)⩾λ2Re 
Q.Theequali y,
λ2Q1+Q2
2=λ2Re Q1+Q2
2, is achie ed i he in eg a ed sys em is no mal.
Secondly, we analyze he supe di usion condi ion by ela ing eigen alues o he He mi ian pa o he
in eg a ed sys em and He mi ian pa o he whole sys em. On one hand, obse ing ha he He mi ian pa
o he in eg a ed sys em coincides wi h he p incipal subma ix o he He mi ian pa o he whole sys em,
we ha e ha λ2Re 
Q⩽λ2Re Q1+Q2
2due o he Cauchy in e lacing heo em o He mi ian ma ices.
On he o he hand, λ2Re 
Q⩾λ2Re Q1+Q2
2+λ2Re Q2−Q1
2due o Ky Fan majo iza ion [26] heo em
λ(A+B)≺λ(A)+λ(B). The e o e, i he di e ence Re (Q2−Q1)2is bounded, i leads o
λ2Re 
Q≈λ2Re Q1+Q2
2.
A supe di usion is hus es ablished
Re λ2(Q)λ2Q1+Q2
2(13)
i (i) he in eg a ed sys em is o close o no mal and (ii) he s uc u e di e ence o coupled laye s
Re Q2−Re Q1is negligibly small.
The equali y o λ2Re 
Q=λ2Re Q1+Q2
2is eached o a mul ilaye ne wo k consis ing o a di ec ed
g aph coupled wi h i s e e sed g aph, Q2=QT
1. In his case, i ollows Re Q2=Re Q1and he e exis s a
supe di usion o Re λ2(Q)>λ
2Q1+Q2
2.
I was epo ed ha he supe di usion occu s when a ully connec ed ne wo k coupled wi h a ne wo k
wi h a ce ain le el o di ec ionali y, quan ified by a me ic called ne wo k di ec ionali y index [16]. He e,
we p o e in appendix Ban open p oblem o he no ma i e p ope y o he ne wo k di ec ionali y index
(NDI). In addi ion, we show ha he condi ion o su ficien di ec ionali y as quan ified by NDI is in line
wi h he de i ed no mali y condi ions in his s udy (as shown in figu e 3). The de i ed condi ion o
supe di usion in his wo k, i.e. a close o no mal and a de ia ion om no mal (uppe le panel in figu e 3)
o he in eg a ed and o e all mul iplex, espec i ely, is equi alen o he condi ion o a su ficien le el o
NDI. The nonno mali y condi ion o he non-supe di usi e egime (lowe igh panel in figu e 3)also
ag ees wi h he NDI condi ion. Howe e , he no no mali y condi ion is widely applicable o mul iplex
ne wo ks wi h any numbe o di ec ed laye s, compa ed o a single di ec ed laye deal by NDI condi ion.
Though he esul s a e demons a ed o wo-laye mul iplex ne wo ks, nume ical esul s on mul iplex
ne wo ks wi h a la ge numbe o laye s show analogous supe di usi e and non-supe di usi e beha io s,
also depending on he nonno mali y and NDI condi ions o he s uc u e o each laye . Howe e , in hose
cases i is a mo e challenging o p o ide a comple e ma hema ical p oo due o he di ficul y in isola ing
he Laplacian o each laye om he block Laplacian.
3.3. Fo mula ion o mul ilaye ne wo ks wi h supe di usion
To in e p e he no mali y condi ions o supe di usion, we p opose a model o cons uc mul iplaye
ne wo ks exhibi ing supe di usion wi h a unable le el o magni ude. When he coupled laye s ha e
iden ical He mi ian pa o Laplacian, he minimiza ion o he no mali y le el o he in eg a ed s uc u e is
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New J. Phys. 23 (2021) 013016 XWange al
Figu e 3. The no mali y condi ion is in line wi h he condi ion o su ficien di ec ionali y o he occu ence o supe di usion.
The size o each ci cle is p opo ional o he alue o ne wo k di ec ionali y index, NDI. The simula ion is pe o med on a
wo-laye ed ne wo k o N=200 nodes whe ein one laye consis s o a ully connec ed, undi ec ed g aph and he o he laye
consis s o a di ec ed g aph whose NDI is modified p og essi ely. The in e coupling weigh is p=200. A high le el o NDI is
ansla ed o a low de ia ion om no mali y o he in eg a ed ne wo k and a high de ia ion o no mali y o he mul iplex
ne wo k as a whole.
Figu e 4. The di e ence Re λ2(Q)−λ2Q1+Q2
2as a unc ion o he in e laye coupling s eng h p o (a) a mul ilaye ne wo k
consis ing o di ec ed ci culan g aph o 100 nodes and a e age deg ee 4 and he cons uc ed laye by he p oposed model; and
(b) a mul ilaye ne wo k consis ing o a gene ic and p o ein in e ac ions ne wo k o he Eps ein–Ba i us ne wo k and he
cons uc ed laye .
ansla ed in o
min Q1QT
2−QT
2Q12(14)
wi h p oo in appendix C. Achie ing supe di usion by minimizing equa ion (14) is in gene al daun ingly
di ficul , e en o undi ec ed g aphs o symme ical Laplacian ma ices. The simul aneous diagonaliza ion,
which always leads o he commu a i i y o ma ices Q1and QT
2,i.e.Q1QT
2=QT
2Q1, was pos ed as an open
p oblem by Hi ia –U u y [27] mo e han a decade ago. Ex ensi e esea ch is pe o med due o he close
associa ion wi h quad a ically cons ained quad a ic p og amming and ce ain ad ances a e made o eal
symme ical ma ices [28]. Howe e , esul s on nonsymme ical ma ices a e ex emely sca ce. None heless,
we p opose a cons uc ion model o gene a e a di ec ed mul ilaye ne wo k by cons uc ing one ma ix (e.g.
Q2) om a gi en ma ix (e.g. Q1), such ha supe di usion occu s.
We decompose he cons uc ion model in o wo pa s: (i) cons uc ing a g aph G2 om he eplica ion
o g aph G1bu a e e e sing he di ec ion o di ec ed links; (ii) p ese ing he ou -deg ee o each node in
g aph G1such ha g aph Laplacian Q2has he same diagonal elemen s as Q1. To ensu e he second smalles
eal pa o he Laplacian eigen alue o be nonze o, we assume a single equi emen o g aph G1 o be a
s ongly connec ed di ec ed g aph. To his end, we p opose o e e se links in a cycle ashion, i.e.
simul aneously e e sing he di ec ion o all links in ol ed in a cycle. In addi ion, he numbe o links
e e sed in a cycle ashion is closely associa ed wi h he magni ude Re λ2(Q)−λ2Q1+Q2
2o
supe di usion. Fo wo di ec ed g aphs G1and G2,wedefinea a iableqas di ec ion o e lap, he ac ion
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New J. Phys. 23 (2021) 013016 XWange al
o links ha ing he same di ec ion and calcula ed by q=|E(G1)∩E(G2)|
|E(G1)∪E(G2)|o in a ma ix o m as
q=i,j(A1)ij(A2)ij
i,j(A1)ij +i,j(A2)ij −i,j(A1)ij(A2)ij
.(15)
The denomina o no malizes he di ec ion o e lap, in which q=0 co esponds o Q2=QT
1and q=1
co esponds o Q1=Q2. The magni ude o supe di usion can hus be uned by e e sing di e en ac ion
o links, pa ame e ized by 1 −p.
Figu e 4shows di usi e beha io s on cons uc ed mul iplex ne wo ks consis ing o (a) a ci culan
di ec ed g aph and he duplica e dig aph wi h 1 −q e e sed links and (b) a eal-wo ld gene ic and p o ein
in e ac ions ne wo k o he Eps ein–Ba i us [29,30] and he duplica e dig aph wi h 1 −q e e sed links.
The coupled dig aph and i s duplica e wi hou link e e sing (q=1) displays a non-supe di usi e
beha io . Fo 0 ⩽q<1, he cons uc ed mul iplex buil upon bo h syn he ic and eal-wo ld di ec ed g aph
shows supe di usion egimes. In addi ion, a highe ac ion o e e sed links (a smalle q) esul sina
highe magni ude o supe di usion.
4. Conclusions
In his pape , we s udy he di usi e dynamics on mul iplex ne wo ks consis ing o a leas one di ec ed
laye s. Though mul ilaye s uc u e o undi ec ed laye s enhances di usion, i is es ic ed o a heo e ical
limi by he agg ega ion o each laye . We show ha ubiqui ous p ope y o di ec ionali y in each laye could
b eak his limi and achie e a unique supe di usion egime, whe ein di usion is e en as e han he
co esponding agg ega ed sys em. We analy ically and nume ically unco e ha he unique supe di usion is
d i en by he di ec ionali y and he unde lying s uc u e o laye s a he han how s ongly laye s a e
coupled. In pa icula , di usi e beha io s a e associa ed wi h he non-no mali y le el o he in eg a ed and
he o e all sys em: when bo h he in eg a ed and he o e all sys em a e no mal, i is assu ed ha he
mul iplex exhibi s only a non-supe di usi e egime; when he in eg a ed sys em is no mal o close o
no mal while he whole sys em is de ia ed om no mal, i exhibi s he unique supe di usi e egime.
Addi ionally, we p o ide a model o cons uc mul iplex ne wo ks, achie ing supe di usion wi h a unable
le el o magni ude. We show ha di ec ionali y induces a unique supe di usion, a egime no exis ed in he
coun e pa o undi ec ed mul ilaye , and sugges he key ole o ne wo k di ec ionali y in analyzing
di usi e dynamics on eal-wo ld sys ems which o en imes a e s uc u ally di ec ed and mul ilaye ed.
Acknowledgmen s
XW acknowledges he p ojec 62003156 suppo ed by NSFC and p ojec ‘PCL Fu u e G ea e -Bay A ea
Ne wo k Facili ies o La ge-scale Expe imen s and Applica ions (LZC0019)’. AT acknowledges pa ial
suppo om NSF G an s EAR-181190 and he UK Resea ch and Inno a ion Global Challenges Resea ch
Fund Li ing Del as Hub G an NES0089261. YM acknowledges pa ial suppo om he Go e nmen o
A ag´
on, Spain h ough a g an o he g oup FENOL (E36-20R), by MINECO and FEDER unds (G an
FIS2017-87519-P) and by In esa Sanpaolo Inno a ion Cen e . The unde s had no ole in s udy design, da a
collec ion and analysis, o p epa a ion o he manusc ip .
Appendix A. Con e gence o he fixed poin
The con e gence o he fixed poin is cha ac e ized by Δx( )T=(x( )−x∗)T.The a eo con e genceis
egula ed by ˙
Δx( )T=d(x( )−x∗)T
d =−(Δx( )T+x∗T)Q=−Δx( )TQ.Theno mΔx( )To he
di e ence ec o is bounded by
Δx( )T=Δx(0)Te−Q ⩽Δx(0)Te−Q .(A.1)
I one mainly conce ns wi h he con e gence beha io o he dec ease o Δx( ), he ollowing holds o
a bounded ope a o Qin a Hilbe space
lim
→∞
log e−Q 
=−λmin (A.2)
whe e λmin is he minimum eigen alue o he Laplacian Q. Fo a connec ed di ec ed g aph, λmin equals o
ze o which means a gua an eed con e gence o he s eady s a e x∗calcula ed by equa ion (3).
8