Applied Ne wo k Science
Cesa i e al. Applied Ne wo k Science (2018) 3:35
h ps://doi.o g/10.1007/s41109-018-0095-y
RESEARCH Open Access
An applica ion o he Shapley alue o he
analysis o co-exp ession ne wo ks
Giulia Cesa i1, Enca nación Algaba2, S e ano Mo e i3* and Juan A. Nepomuceno4
*Co espondence:
[email p o ec ed]
3Uni e si é Pa is-Dauphine, PSL
Resea ch Uni e si y, CNRS,
LAMSADE, 75016 Pa is, F ance
Full lis o au ho in o ma ion is
a ailable a he end o he a icle
Abs ac
We s udy he p oblem o iden i ying ele an genes in a co-exp ession ne wo k using a
(coope a i e) game heo e ic app oach. The Shapley alue o a coope a i e game is
used o asses he ele ance o each gene in in e ac ion wi h he o he s, and o s ess
he ole o nodes in he pe iphe y o a co-exp ession ne wo k o he egula ion o
complex biological pa hways o in e es . An applica ion o he me hod o he analysis
o gene exp ession da a om mic oa ays is p esen ed, as well as a compa ison wi h
classical cen ali y indices. Finally, making u he assump ions abou he a p io i
impo ance o genes, we combine he game heo e ic model wi h o he echniques
om clus e analysis.
Keywo ds: Coope a i e game heo y, Cen ali y, Co-exp ession ne wo ks, Shapley
alue
In oduc ion
Aco-exp ession ne wo k is an undi ec ed g aph whe e he nodes co espond o he genes,
and a link be ween wo genes is es ablished i he wo genes ha e a “simila ” exp ession
p o iles in a da ase (Zhang and Ho a h 2005; Pa migiani e al. 2003; Ma kowe z and
Spang 2007). O e he las wo decades, cen ali y analysis (F eeman 1978;Koschü zki
e al. 2005) was success ully used o measu e he ole played by each gene o in luence
he e y complex sys em o genes’ ela ionships in a co-exp ession ne wo k. Fo ins ance,
some independen wo ks (Be gmann e al. 2004; Ca lson e al. 2006) ha e shown ha in
co-exp ession ne wo ks genes wi h high deg ee-cen ali y a e also likely o be essen ial,i.e.
c i ical o he su i al o di e en o ganisms. In a simila way, in he pape (Gio gi e al.
2013) i was shown ha be weenness cen ali y, ano he measu e o he nodes cen ali y,
is in gene al a posi i e ma ke o essen ial genes in A abidopsis haliana. O he examples
o applica ion o cen ali y measu es o he analysis o gene ic ne wo ks can be ound in
he pape s (Jeong e al. 2001; Junke e al. 2006 and Zampe aki e al. 2010).
On he o he hand, in co-exp ession ne wo ks, genes a e go e ned by complex eg-
ula o y mechanisms and he e ec s on he cell can be app ecia ed only i many genes
simul aneously change hei exp ession beha iou . Fo his eason, i seems aluable o
concei e cen ali y no ions aking in o accoun no only he con ibu ion o single nodes
o he whole s uc u e o he ne wo k, bu also he ole played by each node o all possi-
ble le els o in e ac ion. To his aim, coope a i e game heo y was ecen ly p oposed as
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Cesa i e al. Applied Ne wo k Science (2018) 3:35 Page 2 o 21
a heo e ical amewo k o he design o cen ali y measu es keeping in o accoun he
in e ac ions among genes in subg oups o coali ions. Fo example, ele ance indices based
on coali ional games ha e been success ully applied o di e en kinds o biological ne -
wo ks, such as b ain ne wo ks (Kau man e al. 2005;Keinane al.2004;Kö e e al.2007),
gene ne wo ks (Mo e i e al. 2010), and me abolic ne wo ks (Saji z-He ms ein and
Nikoloski 2012), and o he analysis o di e en biological da a (Saji z-He ms ein and
Nikoloski 2013; Fagnocchi e al. 2015).
In his pape , we apply a game heo e ic index ecen ly in oduced in he pape
(Cesa i e al. 2017) o iden i y he mos ele an genes in a co-exp ession ne wo k.
Such an index gene alizes he no ion o deg ee cen ali y, whose co ela ion wi h
he essen ial genes o di e en biological sys ems is suppo ed by se e al s udies
(see Be gmann e al. (2004), Ca lson e al. (2006), Jeong e al. (2001), Junke e al.
(2006),Zampe aki e al. (2010)). Fi s , we de ine a speci ic coope a i e game, whe e he
playe s a e he genes and he wo h o a se o genes depends on he s uc u e o
he co-exp ession ne wo k and a pa ame e ha speci ies he a p io i impo ance o
each gene. Then, we use he Shapley alue (Shapley 1953) o a coope a i e game o
quan i y he po en ial o a gene in p ese ing he egula o y ac i i y ac oss all pos-
sible subse s o genes in a co-exp ession ne wo k. In he pape (Cesa i e al. 2017)
we used he axioma ic app oach in o de o jus i y he use o he Shapley alue as
a cen ali y measu e, i.e., we p o ed ha he Shapley alue is he unique index ha
sa is y a se o p ope ies wi h a p ecise meaning in he con ex o co-exp ession ne -
wo ks. In his pape , ou objec i e is o show ha he abili y o he Shapley alue
o single ou ele an genes in a co-exp ession ne wo k om he li e a u e, is com-
pa able o he one o o he classical cen ali y measu es. A he same ime, we show
ha he in o ma ion p o ided in e ms o genes selec ed by he Shapley alue is com-
plemen a y o he in o ma ion p o ided by he o he measu es. In o he wo ds, his
pape is de o ed o he applica ion and he alida ion o he ele ance index in o-
duced in he pape (Cesa i e al. 2017), and i s compa ison wi h o he classical cen ali y
indices.
In o de o alida e he use o he ele ance index on a eal gene exp ession da ase
ela ed o lung cance disease (Landi e al. 2008), h ee ele ance analyses a e pe o med,
o di e en choices o he genes’ weigh s: i s , no a p io i knowledge is assumed, i.e.
all genes a e assigned he same weigh ; secondly, a lis o known oncogenes is aken
in o conside a ion by di iding he se o genes in key-genes and non-key-genes and
las ly, he game- heo e ical app oach is combined wi h clus e ing analysis in o de o
assess he ele ance o genes in he ne wo k. A compa ison among he h ee analyses,
as well as a compa ison o Shapley alue o speci ic coali ional games wi h classical
cen ali y indices is p esen ed and he esul s a e in es iga ed om a biological poin
o iew.
The pape is s uc u ed as ollows. “A mo i a ing example” sec ion p esen s a mo i-
a ing example, in o de o cla i y he signi icance and scope o he Shapley alue and
he di e ence wi h espec o classical cen ali y measu es. In “Me hodology”sec ionwe
in oduce he me hodology, desc ibing he game- heo e ical ele ance index and i s in e -
p e a ion as a cen ali y measu e. An applica ion o gene exp ession da a om mic oa ay
echnology is p esen ed in “Expe imen al esul s”and“Conclusions” sec ions. The lis s o
genes selec ed by he Shapley alue a e p o ided as addi ional iles.
Cesa i e al. Applied Ne wo k Science (2018) 3:35 Page 3 o 21
A mo i a ing example
In some cases, classical cen ali y measu es may yield inaccu a e o misleading esul s.
As an example, in he pape (Gai ie i and Sibille 2011) i was shown ha di e en ially
exp essed genes in majo dep ession (i.e. hose genes ha p esen a s a is ically di e en
beha iou in dep essed pa ien s compa ed o heal hy pa ien s) eside in he pe iphe y
o esilien gene co-exp ession ne wo ks, hus sugges ing ha he “cen al” genes a e no
always he mos ele an in he egula o y p ocesses wi hin gene ne wo ks. In he pape
(Ma e al. 2011), he au ho s ha e epo ed he endency o genes wi h highe exp ession
a iance o ha e ewe connec ions ac oss signalling ne wo ks. Mo eo e , in he pape
(Kim e al. 2007), i was obse ed ha p o eins ha ha e been unde posi i e selec ion a e
loca ed a he pe iphe y o he in e ac ion ne wo k.
Fo ins ance, conside he g aph depic ed in Fig. 1. Classical cen ali y measu es
(p ecisely, he deg ee cen ali y (Nieminen 1974;Shaw1954), he closeness cen ali y
(Beauchamp 1965; Sabidussi 1966), he be weenness cen ali y (Ba elas 1948; F eeman
1977)and heei gen ec o cen ali y (Bonacich 1972); see “Classical cen ali y mea-
su es” sec ion o a o mal de ini ion o hese measu es) assign he highes ele ance o
node1.In ac ,node1has hemaximumdeg ee (i.e., numbe o neighbou s), i is he node
wi h he sho es a e age dis ance om all he o he nodes in he g aph (closeness cen-
ali y), i lies on he highes numbe o sho es pa hs connec ing all pai s o o he nodes
(be weenness cen ali y), and i is di ec ly connec ed wi h many cen al nodes (eigen ec-
o cen ali y). On he o he hand, he nodes 2, ..., 6 sha e wo in e es ing cha ac e is ics
ha make hem ele an when he ne wo k depic ed in Fig. 1 ep esen s a co-exp ession
ne wo k:
Fig. 1 A ne wo k wi h 21 nodes
Cesa i e al. Applied Ne wo k Science (2018) 3:35 Page 4 o 21
(1) h ough hei connec ions, hese nodes a e able o in luence he exp ession o all he
o he genes in he ne wo k, i.e. hey in e ac , di ec ly o ia node 1, wi h all he o he
genes wi hin he ne wo k;
(2) he emo al (o inhibi ion) o some o hese nodes b eaks down he egula o y
ac i i y o he ne wo k, by lea ing all he lea nodes isola ed.
In he emaining o his pape , we in oduce and discuss an applica ion o he Shap-
ley alue aimed a measu ing he po en ial o a gene in p ese ing he egula o y ac i i y
wi hin a co-exp ession ne wo k. On a co-exp ession ne wo k buil o e a da ase om
he li e a u e, we show ha he Shapley alue o he coali ional game in oduced in he
pape (Cesa i e al. 2017) can be in e p e ed in e ms o he abili y o genes o abso b
he e ec s o he inhibi ion o o he co ela ed genes. S a ed di e en ly, we show ha he
Shapley alue highligh s he ole o genes in he o e all “connec i i y” o a co-exp ession
ne wo k, by aking in o accoun he e ec ha hei emo al has o e he induced sub-
ne wo ks. In his sense, in Fig. 1, node 2 (as well as nodes 3, 4, 5 and 6) is mo e ele an
han node 1: when node 1 is emo ed, he ne wo k is di ided in o i e componen s, whose
o e all egula ion is main ained hanks o he p esence o nodes 2, 3, 4, 5 and 6 espec-
i ely. On he o he hand, when one o hese las nodes is emo ed, he ne wo k is spli
in ou componen , h ee o which a e no longe able (as being isola ed nodes) o main-
ain hei egula o y ac i i y. Wi h he objec i e o p o ide an index aimed a ep esen ing
his ype o ele ance o genes in a co-exp ession ne wo k, in he ollowing we conside a
coali ional game whe e he alue o a coali ion o genes depends on he ca dinali y o he
coali ion i sel and o i s neighbou hood. The mo e he genes ha a e di ec ly in e ac ing
in he ne wo k wi h genes in he coali ion, and he e o e he abili y o he coali ion o keep
he ne wo k connec ed, he highe he s eng h o he coali ion. Following he app oach
in oduced in he pape (Cesa i e al. 2017), we p opose he Shapley alue o such a coali-
ional game as a ele ance index o genes in co-exp ession ne wo ks, aking in o accoun
he ma ginal con ibu ions o genes o he connec i i y o all he coali ions o genes in he
ne wo k. We use he Shapley alue o assess he ele ance o genes in a eal co-exp ession
ne wo k ela ed o lung cance , by means o h ee di e en analyses. On such a ne wo k,
when no a p io i knowledge is assumed abou he genes unde analysis (see, o ins ance,
he i s analysis in “Fi s analysis” sec ion), he Shapley alue is able o highligh he
ole o genes in he o e all connec i i y o he ne wo k, by assigning he highes ele-
ance o hose genes ha sha e he wo a o emen ioned cha ac e is ics. We a gue ha
his in e es ing beha iou o he Shapley alue o single ou nodes ha may b eak down
he egula o y ac i i y o he ne wo k holds in gene al o spa se g aphs cha ac e ized by
a ela i e low numbe o cycles; whe eas in g aphs whe e he pe iphe al nodes belong
o mo e connec ed componen s o clus e s, he indica ion p o ided by a high Shapley
alue seems mo e ela ed o he ole o ce ain genes o media e he egula ion be ween
a clus e and he o he s uc u es o he ne wo k ( his poin will be u he discussed in
Example 2).
Me hodology
Classical cen ali y measu es
An undi ec ed g aph o ne wo k is a pai N,E,whe eNis a ini e se o e ices o nodes
and Eis a se o edges eo he o m {i,j}wi h i,j∈N,i= j.
Cesa i e al. Applied Ne wo k Science (2018) 3:35 Page 5 o 21
We de ine he se o neighbou s o a node iin g aph N,Eas he se Ni(E)={j∈N:
{i,j}∈E},and hedeg ee o ias he numbe di(E)=|Ni(E)|o neighbou s o iin g aph
N,E. Wi h a sligh abuse o no a ion, we deno e by NS(E)={j∈N:∃i∈Ss. .j∈Ni(E)}
he se o neighbou s o nodes in S∈2N,S=∅,andin heg aphN,E.Apa h be ween
nodes iand jin a g aph N,Eis a ini e sequence o nodes (i0,i1, ..., ik),whe ei=i0and
j=ik,k≥1, such ha {is,is+1}∈E o each s∈{0, ··· ,k−1}and such ha all hese
edges a e dis inc . Two nodes i,j∈Na e connec ed in N,Ei i=jo i he e exis s a
pa h be ween iand jin E.Theleng h o a pa h be ween iand jis henumbe o edges
in he pa h and a sho es pa h be ween iand jis a pa h be ween iand jwi h minimum
leng h. Le i∈Nand S⊆N {i}.
Cen ali y measu es assign o each node in a ne wo k a alue ha co esponds o some
ex en o he ele ance o ha node wi hin he ne wo k s uc u e. The ou classical
cen ali y measu es conside ed in his pape a e he ollowing:
(1)
Deg ee cen ali y
(Nieminen 1974;Shaw1954): he deg ee cen ali y o i∈Nis
de ined as |Ni(E)|, i.e. he numbe o neighbou s o
i
in g aph N,E.I isanindexo
he po en ial communica ion ac i i y o a node.
(2)
Closeness cen ali y
(Beauchamp 1965; Sabidussi 1966): he closeness cen ali y o
node
i
is de ined as |N|−1
j∈Nh(i,j), whe e h(i,j)is he dis ance be ween
i
e
j
, i.e. he leng h
o he sho es pa h be ween
i
and
j
. I measu es o wha ex en a node can a oid he
con ol po en ial o he o he s nodes.
(3)
Be weeness cen ali y
(Ba elas 1948; F eeman 1977): he be weenness cen ali y o a
node
k
is de ined as i,j∈Nbij(k), whe e bij(k)=gij(k)
gij and gij is he numbe o
sho es pa hs be ween nodes
i
and
j
,whilegij(k)is he numbe o sho es pa hs
be ween nodes
i
and
j
ha con ain
k
. I is an index o he po en ial o a node o
con ol o communica ion.
(4)
Eigen ec o cen ali y
(Bonacich 1972): he eigen ec o cen ali y o a node
i
is
de ined as he i− h elemen o he p incipal eigen ec o o he adjacency ma ix
A=aijco esponding o N,E, whe e aij =1i {i,j}∈Eand aij =0o he wise. I
assigns high cen ali y o nodes ha a e highly connec ed o nodes who hemsel es
ha e high cen ali y.
A game- heo e ic ele ance index
Le N,Ebe a co-exp ession ne wo k, ha is a ne wo k whe e he se o nodes N ep-
esen s a se o genes and he se o edges Edesc ibes he in e ac ion among genes, i.e.
he e exis s an edge be ween wo genes i hey a e di ec ly in e ac ing in he biological
condi ion unde analysis. Mo eo e , le k∈RNbe a pa ame e ec o ha speci ies he a
p io i impo ance o weigh o each gene. Acco ding o Cesa i e al. (2017), we de ine he
coali ional game N, k
E,whe eNis he se o genes unde s udy and he cha ac e is ic
unc ion k
Eassigns a wo h o each coali ion o genes S⊆N ep esen ing he o e all mag-
ni ude o he in e ac ion be ween he genes in S, which akes in o accoun he weigh (i.e.,
he a p io i impo ance) o each gene di ec ly connec ed o Sin he biological ne wo k.
Mo e p ecisely, he map k
E:2
N→Rassigns o each coali ion S∈2N {∅} he alue
k
E(S)=
j∈S∪NS(E)
kj(1)

Cesa i e al. Applied Ne wo k Science (2018) 3:35 Page 6 o 21
ha is he sum o he weigh s associa ed o he genes in Sand o he ones ha a e di ec ly
connec ed in N,E o some genes in S(by con en ion, k
E(∅)=0) (Cesa i e al. 2017). The
class o games (N, )de ined acco ding o ela ion (1), on some gene ne wo k G≡V,E
and wi h pa ame e k∈RN, is deno ed by EKN.
A well-known solu ion o coali ional games is he Shapley alue (Shapley 1953), which
was in oduced in 1953 and since hen applied o a wide ange o ields, including biology
(Mo e i and Pa one 2008). The Shapley alue ρ( )o a game (N, )is de ined as he
a e age o ma ginal ec o s o e all |N|! possible o de s in N(|N|is he ca dinali y o
he se N). In o mula
ρi( )=
σ∈N
mσ
i( )
|N|! o alli∈N,(2)
whe e Nis he se o all possible pe mu a ions o he elemen s in Nand σ(i)=jmeans
ha wi h espec o σplaye iis in he j- h posi ion, and whe e he ma ginal ec o
mσ( )∈RNis de ined by mσ
i( )= ({j∈N:σ(j)≤σ(i)})− ({j∈N:σ(j)<σ(i)}) o
each i∈N(i.e., mσ
i( )is he ma ginal con ibu ion o playe i o he coali ion o playe s
wi h lowe posi ions in σ).
In he pape (Cesa i e al. 2017), he au ho s ha e shown ha he Shapley alue is he
unique ele ance index o genes (de ined as a map ρ:EKN→RN) which sa is ies
ou desi ed p ope ies, namely, symme y, he dummy playe p ope y, e iciency and
s a -addi i i y. Th ee ou o hese ou p ope ies a e na u al axioms bo owed om he
ela ed li e a u e on coope a i e games: he p ope y o symme y equi es ha i wo
genes iand jha e he same a p io i weigh ki=kjand in addi ion, hey a e connec ed
o he same se o neighbou s in a ne wo k, hen hey should ha e he same ele ance;
he dummy playe p ope y basically implies ha he ele ance o a disconnec ed node
in a ne wo k coincides wi h i s a p io i impo ance; he e iciency axiom de e mines
he scale o measu e, se ing he sum o he ele ance o all genes equal o he sum o
hei a p io i weigh s. The ou h p ope y in oduced in he pape (Cesa i e al. 2017)
o axioma ically cha ac e ize he Shapley alue on he class EKN, i.e., he s a -addi i i y
axiom, says ha inc easing he a p io i weigh o a node i om 0 o a posi i e alue
should a ec he ele ance o gene iand i s neighbou s a he same ex en , o wha e e
g aph. Consequen ly, ealloca ing he a p io i impo ance o a node among i s neighbou s,
he s a addi i i y p ope y ca ches he idea o measu ing he abili y o nodes o abso b
he changes in exp ession o co ela ed genes, as p e iously discussed in he mo i a ing
example illus a ed in “A mo i a ing example”sec ion.
A p ac ical limi a ion inhe en o mos o he applica ions o he Shapley alue is he
compu a ional bu den ela ed o i s calculus. Su p isingly, in he pape (Cesa i e al. 2017)
he au ho s ha e shown ha he Shapley alue o a coali ional game N, k
Ecan be
compu ed acco ding o he ollowing much simple ela ion:
ρi k
E=
j∈(Ni(E)∪{i})
kj
dj(E)+1,(3)
o each i∈N.
Acco ding o ela ion (3), a gene connec ed o many genes who hemsel es ha e a low
deg ee ge s a high Shapley alue (in o he wo ds, he ele ance o a gene inc eases wi h he
numbe o i s neighbou s ha ing a low deg ee). Rela ion (3) also sugges s ha genes wi h a
Cesa i e al. Applied Ne wo k Science (2018) 3:35 Page 7 o 21
high Shapley alue would be able o in e ac di ec ly wi h he maximum numbe o o he
nodes in he ne wo k and i s emo al would spli he ne wo k in a maximum numbe o
connec ed componen s wi h ew genes, o e en ually cons i u ed by isola ed genes. As
shown in he pape (Aadi hya 2010), ela ion (3) can be calcula ed ia an O(|N|+|E|)
p ocedu e, which makes possible i s compu a ion on e y la ge ne wo ks ( ecall ha on
ane wo kN,E, calcula ing be weenness cen ali y akes O(|N||E|) ime using B andes’
algo i hm (B andes 2001). We conclude his sec ion showing he esul s o ela ion (3)
applied o he mo i a ing example in “A mo i a ing example”sec ion.
Example 1 Conside he gene ne wo k in Fig. 1. Suppose all he genes ha e he same a
p io i weigh ki=1∀i∈N. Then, by ela ion (3)ρ k
E=35
30 ,56
30 ,56
30 ,56
30 ,56
30 ,56
30 ,21
30 ,21
30 ,
21
30 ,21
30 ,21
30 ,21
30 ,21
30 ,21
30 ,21
30 ,21
30 ,21
30 ,21
30 ,21
30 ,21
30 ,21
30 . The e o e, he Shapley alue gi es he high-
es ele ance o nodes 2,3,4,5 and 6, ollowed by node 1 and he leas ele ance o he lea
nodes {7, ...,21}. Ins ead, all he o he classical cen ali y measu es de ined in “Classical
cen ali y measu es” sec ion p o ide he ollowing anking: node 1 is anked i s , ollowed
by nodes {2, 3, 4, 5, 6}in he second posi ion and, inally, by he lea nodes wi h he lowes
ank.
Example 2 Now, conside he gene ne wo k in Fig. 2. Suppose again ha all he genes
ha e he same a p io i weigh ki=1∀i∈N.AsinExample1, he middle posi ion o node 1
in he g aph can lead o he conclusion ha 1 is he mos cen al gene, a leas , i we adop a
no ion o impo ance ela ed o he idea ha genes in luence each o he s ia sho es pa hs.
Fig. 2 A ne wo k wi h 13 nodes and h ee cliques wi h ou nodes each
Cesa i e al. Applied Ne wo k Science (2018) 3:35 Page 8 o 21
In ac , in he ne wo k o Fig. 2, bo h he closeness cen ali y and he be weenness cen ali y
ank node 1 in he highes posi ion. On he o he hand, i we a e in e es ed in measu ing
he impo ance o nodes in ela ion o hei abili y o in luence hei neighbou s, wi hou
any a p io i assump ion on who is in luenced by whom, i seems easonable o conside
nodes 2, 3 and 4 as he mos cen al ones, because o hei ole o connec each clique o
ou node (i.e., each subse o ou nodes such ha e e y wo dis inc e ices in he subjec
a e adjacen ) wi h he o he pa s o he ne wo k. So, i is no su p ising ha he deg ee
cen ali y, he eigen ec o cen ali y and he Shapley alue gi e he highes ele ance o such
nodes. This example emphasizes he ac ha di e gen bu s ill easonable conclusions
can be ob ained as a logical consequence o al e na i e no ions o ele ance o nodes o
a ne wo k. Howe e , we ecognize ha he in e p e a ion o he Shapley alue gi en in he
mo i a ing example o “A mo i a ing example” sec ion does no apply o he g aph o Fig. 2,
which is cha ac e ized by mul iple cliques: he emo al o one o he nodes 2, 3 and 4 does
no isola e any lea node, bu only ully connec ed componen s. No e also ha he Shapley
alue is he unique measu e, among he ones conside ed in Table 1, which anks nodes in
he cliques s ic ly highe han node 1 (only he deg ee cen ali y assigns he same alue
equal o 3 o he nodes in he cliques and o node 1). In ac , acco ding o he Shapley
alue, he nodes in a clique ga he some u he ele ance om being connec ed o each
o he , and hen om being less exposed o he in luence o o he nodes. On he con a y,
node 1 is di ec ly connec ed o he e y well connec ed nodes 2, 3 and 4, and i is exposed
o hei in luence. On he opposi e side, he be weenness cen ali y anks node 1 in he
op, and he nodes in he cliques in he bo om: we a gue ha his nega i e co ela ion
be ween he Shapley alue and he be weenness cen ali y is also due o he ela i e high
numbe o cliques p esen in he ne wo k o Fig. 2, which is no he case o he o he ne wo ks
conside ed in his pape .
Rela ed li e a u e
Ano he way o keep in o accoun he a p io i impo ance o genes was p oposed in he
pape (Mo e i e al. 2010) by means o he so-called associa ion game,whe ease o key-
genes K⊂N(e.g. a se o genes known a p io i o be in ol ed in biological pa hways
ela ed o ch omosome damage) is conside ed and he alue assigned o a coali ion Sis
he numbe o key-genes in e ac ing only wi h S o mally, in he pape (Mo e i e al 2010)
he alue assigned o a coali ion S⊆Nis he ca dinali y o he se {i∈K:Ni(E)⊆S}.
Howe e , he de ini ion p oposed in ela ion (1) seems mo e lexible o explo e all pos-
sibili ies o ecip ocal in luence among genes. I gene alizes he game in oduced in he
pape s (Su i and Na aha i 2008; Aadi hya e al. 2010) o de e mining he “ op-knodes”
in a co-au ho ship ne wo k, by he in oduc ion o a pa ame e ha speci ies he a p i-
o i impo ance o each node. The pa ame e ec o kallows o an a p io i anking o he
Table 1 Compa ison o cen ali y measu es in he ne wo k o Example 2
12345678910111213
ρ0.85 1.20 1.20 1.20 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95
Deg ee3444333333333
Closeness 0.048 0.038 0.038 0.038 0.029 0.029 0.029 0.029 0.029 0.029 0.029 0.029 0.029
Be weenness 48 27 27 27 0 0 0 0 0 0 0 0 0
Eigen ec o 0.92 1 1 1 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.79
Cesa i e al. Applied Ne wo k Science (2018) 3:35 Page 9 o 21
genes acco ding o hei impo ance, while in he p e ious model in oduced in he pape
(Mo e i e al. 2010) only a wo-le el dis inc ion was made be ween key-genes and non
key-genes. Mo eo e , by measu ing o wha ex en a coali ion o genes is connec ed o he
es o he ne wo k, ela ion (1) gene alizes he no ion o deg ee cen ali y o g oups o
genes, which is jus i ied by some p ac ical e idences showing a s ong co ela ion be ween
he deg ee cen ali y and genes ha a e essen ial o di e en biological unc ions (see, o
ins ance, (Be gmann e al. 2004; Ca lson e al. 2006;Jeonge al.2001; Junke e al. 2006;
Zampe aki e al. 2010). In ac , i only he weigh o genes inside a coali ion was o be con-
side ed (and no he one o he neighbou s, as in ou de ini ion), he cen ali y measu e
ob ained h ough ela ion (3) would coincide wi h a “weigh ed” deg ee cen ali y.
Expe imen al esul s
This sec ion is de o ed o he analysis o he Shapley alue on co-exp ession ne wo ks
gene a ed om a gene exp ession da ase . We i s in oduce and discuss a p elimina y
analysis o he obus ness o he me hodology based on he Shapley alue in selec ing he
mos ele an genes.
In he ollowing, he c i e ion used o es ablish whe he wo genes a e co-exp essed
is based on he co ela ion be ween hei exp ession p o iles in he co esponding gene
exp ession da ase (p ecisely, on he Pea son’s co ela ion coe icien ). Basically, wo genes
a e said co-exp essed i and only i hei Pea son’s co ela ion coe icien is la ge han a
p ede ined cu -o (Ca e e al. 2004; Zhang and Ho a h 2005). O cou se, he choice
o he h eshold is c i ical o he analysis. The e o e, we s a wi h he e alua ion o he
obus ness o he model using al e na i e h esholds.
Robus ness e alua ion
We es ed he model on a andomly gene a ed symme ic ma ix o size 1000 wi h en ies
in he ange [ 0, 1]. To be mo e speci ic, we used a ma ix whe e he elemen in ow iand
column j ep esen s he co ela ion be ween gene iand gene jin a ic i ious, andomly
d awn da ase o 1000 genes. Fo he sake o his analysis he pa ame e ec o kwas ixed
in such a way ha ki=1 o e e y i. The ma ix was ans o med in a boolean adjacency
ma ix (whe e 1 ep esen s a connec ion in he ne wo k and 0 means no connec ion)
acco ding o h ee di e en h esholds, 0.7, 0.8 and 0.9, espec i ely. A ne wo k was gen-
e a ed o each h eshold acco ding o he a o emen ioned c i e ion and he ele ance
index o each gene, i.e. he Shapley alue ρo he game de ined in (1), was compu ed
ia ela ion (3). A compa ison be ween he esul s o he h ee di e en h esholds was
conduc ed. In pa icula , we selec ed he lis o he 5% o genes wi h he highes Shapley
alue o each h eshold, and we ob ained he ollowing esul s: 18 genes a e commonly
selec ed by he Shapley alue o cu o 0.7 and 0.8; 15 genes a e commonly selec ed o
cu o 0.8 and 0.9 and 5 genes a e commonly selec ed o cu o 0.7 and 0.9 These esul s
a e summa ized in Table 2.
Table 2 Numbe o common genes by using a cu o o 0.7, 0.8 and 0.9, espec i ely
0.7 0.8 0.9
0.7 50 18 5
0.8 18 50 15
0.9 5 15 50
Cesa i e al. Applied Ne wo k Science (2018) 3:35 Page 16 o 21
Fig. 9 Compa ison o he ela ion among he deg ee o a node and he deg ee o i s neighbou s. The poin s
ep esen genes and hei coo dina es a e gi en, espec i ely, by he deg ee o a gene (on he x-axis) and he
mean deg ee o i s neighbou s (on he y-axis). The colou ed poin s ep esen he genes selec ed by he
di e en cen ali y measu es
wi h a Cy oscape plug-in called Agilen Li e a u e Sea ch (Sai o e al. 2012). Second, we
also pe o med a Reac ome s udy wi h he same goal. I is impo an o no e ha only he
i s 100 genes o each analysis in he Addi ional ile 1: Table S1, Addi ional ile 2:TableS2
and Addi ional ile 3: Table S3 ha e been s udied due o he limi a ions o hese ools.
The Cy oscape plug-in sea ches a se o genes in published pape s a ailable in public
eposi o ies such as PubMed. The sea ch has been pe o med by aking as inpu he lis o
genes selec ed by he Shapley alue and a se o key-wo ds, namely “Homo sapiens” and
“Adenoca cinoma”. The ool p o ides as a esul he subse o genes ha a e ci ed in he
ela ed li e a u e.
Figu e 10 shows on he le -hand side he esul s o he Li e a u e Mining-based com-
pa ison. The i s , second and hi d analysis epo 70, 57 and 62 genes ha a e ci ed in
he li e a u e, espec i ely. The i s analysis seems o epo mo e known genes bu he
h ee analyses ob ain compa able esul s, by inding in he li e a u e mo e han a hal o
he genes selec ed by he Shapley alue.
Mo eo e , a s udy based on Reac ome (C o and e al. 2004) was pe o med in o de
o compa e he h ee analyses. Reac ome is a eposi o y o biological pa hways, namely
g oups o eac ions among nucleic acids, p o eins and ano he kind o molecules ha
in e ac as pa o biological p ocesses as o example he egula ion o gene exp ession,
me abolism, e c. The h ee lis s o 100 genes ha e been analyzed, yielding he ollowing
esul s: he i s analysis iden i ies 51 genes, he second 45 and he hi d 47 (see he igh -
hand side o Fig. 10). These esul s a e cohe en wi h he Li e a u e Mining-based esul s.

Cesa i e al. Applied Ne wo k Science (2018) 3:35 Page 17 o 21
Fig. 10 Compa ison o h ee analysis based on Li e a u e Mining and Reac ome s udy
Mo eo e , we obse e ha he i s analysis epo s 329 pa hways, he second 379 and
he hi d 219. This in o ma ion could indica e ha he quali y o he genes ound by he
second analysis is highe ha he o he wo analyses.
The Ne wo k Cance o Genes ool (NCG5.0) was used o u he in es iga e he esul s
o he analysis om a biological poin o iew. This ool only p o ides in o ma ion abou
known cance genes and i is he e o e oo es ic i e o be used in a quan i a i e com-
pa ison as be o e. Fo example, a gene could be ele an as ac ing as a “swi ch” o a known
oncogene (cance gene) o co- egula e an impo an ela ed p ocess bu i would no be
epo ed by NGC, unless i is i sel an oncogene. Howe e , his ool p o ides some use-
ul in o ma ion om a quali a i e poin o iew, allowing us o e alua e he esul s o ou
analysis and o compa e hem on he basis o he in o ma ion i p o ides.
The i s 100 genes o each analysis in Table 3ha e been s udied wi h NCG, wi h he
objec i e o unde s anding hei biological ele ance om a quali a i e pe spec i e. The
i s analysis inds 5 oncogenes, he second 20 and he hi d 11. These esul s suppo he
idea ha he second analysis epo s genes wi h a highe quali y. Howe e , i is impo an
o emphasize ha he second analysis uses a p io i in o ma ion, by conside ing as inpu
23 well-known lung cance genes, p ecisely ob ained using NCG. I mus be no ed ha 15
genes ou o hese 20 we e used as inpu key-genes. The e o e, we a gue ha each analysis
iden i ies, espec i ely, 5, 5 and 11 no p e iously known oncogenes. The i s and he
hi d analysis do no use any a p io i knowledge. Ne e heless, 4 ou o 5 genes ob ained
by he i s analysis a e in he well-known se o 23 lung cance genes, as well as 2 ou o
11 genes ob ained by he hi d analysis.
Mo eo e , i is in e es ing o u he in es iga e hose genes ha a e epo ed only by
he p oposed ele ance index bu no by he o he (classical) cen ali y measu es. Wi h
espec o his, he i s analysis p esen s 19 genes, he second 71 and he hi d 33 ha
a e selec ed only by he Shapley alue. These se s o genes ha e also been analyzed using
he NCG ool. The i s analysis does no show any known cance gene acco ding o he
in o ma ion suppo ed by he ool. Howe e , he second analysis epo s 16 o 71 genes
Cesa i e al. Applied Ne wo k Science (2018) 3:35 Page 18 o 21
as cance genes and he hi d analysis 3 o 33. So, he second analysis p esen s he bes
esul s in his sense, bu i mus be no ed ha 15 o he 16 genes a e p ecisely pa o he
23 lung cance genes used as a p io i in o ma ion in he second analysis. The e o e, he
second analysis only epo s 1 cance gene which is no p e iously known and used as
inpu .
The cance genes epo ed by he second analysis ha belongs o he se o key-
genes used as inpu a e ATXN3L,CDH10,COL11A1,DACH1,DNAH3,FGFR4,GRM8,
HLA-A,NRAS,PAK3, PDIA4, PPP1R3A,PTPRD,RUNX1T1, and ZMYND10. The gene
G6PC is also epo ed by he second analysis bu i is no included in he inpu se o
key genes. This gene is a li e cance gene wi h a unc ionali y ela ed o he egula ion
o in acellula p ocesses and me abolism. Fu he mo e, Table 5shows he cance genes
epo ed by he hi d analysis. I can be obse ed ha hey a e leukemia, lung and glioblas-
oma cance genes. The gene CD1Bis a lung cance ha belongs o he se o key genes
used in he second analysis. I is impo an o no e ha his a p io i in o ma ion is no
used in he hi d analysis.
Conclusions
In his pape , we p oposed a ele ance index o nodes in gene co-exp ession ne wo ks,
wi h he objec i e o measu ing he po en ial o genes in ac ing as in e media ies be ween
hub nodes and lea nodes and p ese ing he egula o y ac i i y wi hin gene ne wo ks. Fo
his pu pose, we used a game- heo e ic app oach, by de ining a coope a i e game whe e
he s eng h o a coali ion o genes depends on he a p io i impo ance o he genes in i s
neighbo hood. The Shapley alue o such a game is p oposed as a new ele ance index
o genes. Ou me hodology is suppo ed by a p ope y-d i en app oach, whe e he se
o p ope ies sa is ied by he Shapley alue ha e a biological in e p e a ion. Mo eo e , an
expe imen al s udy is conduc ed on a gene exp ession da ase om mic oa ay echnol-
ogy, ela ed o a lung cance disease and he esul s o he Shapley alue a e compa ed
wi h classical cen ali y measu es.
The e sa ili y o he ele ance index and i s e y low compu a ional complexi y
(O(|N|+|E|) allow he combina ion o a game- heo e ical app oach wi h o he echniques
om ne wo k analysis. Indeed, we used an algo i hm om clus e analysis ha iden i ies
o e lapping clus e s o genes, in o de o assess he a p io i impo ance o genes in he
ne wo k unde analysis. An in e es ing di ec ion o u u e esea ch is he u he s udy
o hese echniques, in o de o e ine he ele ance analysis, and he applica ion o ou
model o o he gene ne wo ks in o de o p o ide new biological knowledge.
Table 5 Thi d analysis: cance genes epo ed only by he Shapley alue (and no by o he cen ali y
measu es)
Gene P o ein unc ion P ope ies P ima y Cance
name si e ype
GNAT1 Cell esponse o s imuli/signal
ansduc ion
In e ac ion wi h 5 p o eins and i ’s
pa o a complex
Blood Leukemia
CD1B This gene has no unc ional
in o ma ion
In e ac ion wi h 2 p o eins and i ’s
pa o a complex
Lung Lung
GML Cell cycle/ egula ion o in acellula
p ocesses and me abolism/signal
ansduc ion
In e ac ion wi h p o eins B ain Glioblas oma
Cesa i e al. Applied Ne wo k Science (2018) 3:35 Page 19 o 21
We wan o emphasize he e ha we canno expec ha a single ele ance index, cha ac-
e ized by a so low complexi y, could cap u e all possible c i ical aspec s o he p oblem.
As i o en happens in he p ope y-d i en analysis o cen ali y measu es, na u al p op-
e ies sa is ied by an index in a gi en class o g aphs may be ou weighed by a less in ui i e
p ope y o he same index in a di e en amily o g aphs. This is he case, o ins ance,
o dense g aphs o g aphs wi h a ele an numbe o cliques, as he one conside ed in
Example 2. E en i he in e p e a ion o he anking o nodes o Example 2is cohe en
wi h an e alua ion o he leade nodes in a communi y s uc u e (Li and Daniels 2015;
Li e al. 2015,2018), he in e p e a ion o he Shapley alue along he lines discussed in
he mo i a ing example o “A mo i a ing example” sec ion and in Example 1,doesno
apply o he ne wo k o Fig. 2. E en i many biological ne wo ks, such as he co-exp ession
ne wo ks examined in his wo k ( ecall ha , as desc ibed in “Desc ip ion o he da ase ”
sec ion, a ne wo k is cha ac e ized by an equilib ium be ween connec i i y and spa si ica-
ion (Ch is iano Sil a and Zhao 2016)), con ain mo e subs uc u e esembling he g aph
o Fig. 1 han esembling he one o Fig. 2, a p ecise cha ac e iza ion o he class o g aphs
whe e he in e p e a ion o he Shapley alue p o ided in “A mo i a ing example”sec ion
applies, is s ill an open ques ion and an in e es ing issue o u u e esea ch.
Addi ional iles
Addi ional ile 1:Table S1: Genes selec ed by ρ( i s analysis) (PDF 65 kb)
Addi ional ile 2:Table S2: Genes selec ed by ρ(second analysis) (PDF 54 kb)
Addi ional ile 3:Table S3: Genes selec ed by ρ( hi d analysis) (PDF 64 kb)
Abb e ia ions
GEO: Gene exp ession omnibus; NCBI: Na ional cen e o bio echnology in o ma ion; NCG: Ne wo k cance o genes
Acknowledgemen s
We hank h ee anonymous e e ees o hei aluable sugges ions and commen s on a o me e sion o his pape .
Au ho s’ con ibu ions
GC, EA and SM de eloped he game heo e ic model; GC and JAN p ocessed he da a and pe o med he analysis. All
au ho s w o e, ead, and app o ed he manusc ip .
Compe ing in e es s
The au ho s decla e ha hey ha e no compe ing in e es s.
Publishe ’s No e
Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps and ins i u ional a ilia ions.
Au ho de ails
1Depa men o Ma hema ics, Poli ecnico di Milano, Milano, I aly . 2Depa men o Applied Ma hema ics and IMUS,
Uni e si y o Se ille, Se ille, Spain . 3Uni e si é Pa is-Dauphine, PSL Resea ch Uni e si y, CNRS, LAMSADE, 75016 Pa is,
F ance . 4Depa men o Compu e Languages and Sys ems, Uni e si y o Se ille, Se ille, Spain .
Recei ed: 20 Feb ua y 2018 Accep ed: 14 Augus 2018
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