A dynamical approach to identify vertices centrality in complex networks

A

dynamical

approach

to

identify

vertices’

centrality

in

complex

networks

Long Guo

a

,

b

,

∗

,

Wen-Yao Zhang

b

,

Zhong-Jie Luo

a

,

Fu-Juan Gao

a

,

Yi-Cheng Zhang

c

,

b

,

∗

a_{SchoolofMathematicsandPhysics,ChinaUniversityofGeosciences(Wuhan),430074,Wuhan,China} b_{DepartmentofPhysics,UniversityofFribourg,CH1700,Fribourg,Switzerland}

c_{InstituteofGreenandIntelligentTechnology,ChineseAcademyofSciences,400714,Chongqing,China}

Keywords: Dynamicalcentrality DSA

TheKuramoto model Complexnetwork

In this paper, we proposed a dynamical approach to assess vertices’ centrality according to the synchronization process of the Kuramoto model. In our approach, the vertices’ dynamical centrality is calculated based onthe Difference ofvertices’ Synchronization Abilities(DSA), whichare different fromtraditionalcentralitymeasurementsthatarerelatedtothetopologicalproperties.Throughapplying our approachtocomplexnetworks withaclearcommunity structure,wehave calculatedall vertices’ dynamicalcentralityandfoundthatverticesattheendofweaklinkshavehigherdynamicalcentrality. Meanwhile,weanalyzed therobustnessandefficiency ofourdynamical approachthroughtesting the probabilitiesthatsomeknownvitalverticeswererecognized.Finally,weappliedourdynamicalapproach toidentifycommunityduetoitssatisfactoryperformanceinassessingoverlappingvertices.Ourpresent workprovidesanewperspectiveandtoolstounderstandthecrucialroleofheterogeneityinrevealing theinterplaybetweenthedynamicsandstructureofcomplexnetworks.

1. Introduction

Recentdecadeshavewitnesseda vigorousdevelopmentofthe network science, which is an interdisciplinary academic field to understand the behavior of natural, social andtechnical systems underthe fundamental framework ofcomplex networks[1]. Em-pirical analysis shows that many real complex networks, where vertices represent the elementary units of a given system and links describe the interactions betweenunits [2,3], exhibit some nontrivial properties, such as the heterogeneous nature of ver-tices(e.g.,thepower-lawdistributionofvertex’sconnectivity)and thecommunitystructure(alsocalledclustering ormodule).Those propertiesdemonstratethatheterogeneitymeansdifference,which indicatesthat toidentifythevitalverticeshasitsremarkablerole inanalyzingthestructureanddynamicsofcomplexnetworks[4]. Forexample, incomplex networkwith aclear community struc-ture, vital verticeswithhigher betweennesscentrality, whichare called the overlapping vertices, belong to more than one

com-*

Correspondingauthors.LongGuoat:SchoolofMathematicsandPhysics,China UniversityofGeosciences(Wuhan),430074,Wuhan,China;Yi-ChengZhangat: In-stituteofGreenandIntelligentTechnology,ChineseAcademyofSciences,400714, Chongqing,China.

E-mailaddresses:guolong@cug.edu.cn(L. Guo),yi-cheng.zhang@unifr.ch (Y.-C. Zhang).

munity commonly.Toidentifytheoverlapping verticeseffectively benefitsthecommunitydetection[5–8].Meanwhile,theinfluential verticescanbequantifiedbyvariousindexes.Takedegree central-ityforexample,verticeswithlarger degreehavean abilityto in-fluencemoreothervertices.Tomonitorthoseinfluentialverticesis helpfulforthepredictionandcontrolofspreadingdynamics[9,10]. The vital vertices, which can be identified using the concept of centrality, arelargely affected andreflected by the topological structureanddynamicalpatternofthenetworktowhichthey be-long. A tremendous numberof methods forcentralityhave been proposed and well studied mainly based on the local or global topological structure [4,11],such asthe degree centrality, K-core decomposition, betweennesscentralityandeigenvector centrality. Andthosemethodsneglectthecriticalroleofdynamicalprocesses inidentifyingvertices’importance.Althoughsomedynamics-based centrality has been studied [4,9,12–14], to identify the vital ver-ticeseffectivelyandefficientlyfromthedynamicalperspectivealso remains a big challenge. Since the structure and dynamics are tightly coupledincomplexnetworks,dynamicsisfundamental in assessing the impact of vertices in global performance [15–18]. Forexample,theKuramotomodelsharesdifferentsynchronization behaviors on the homogeneous and heterogeneous complex net-works[19].

As iswell known,synchronizationis acollective phenomenon occurring in systems of mutual interaction between units and

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Published in "Physics Letters A 381 (48): 3972–3977, 2017"

which should be cited to refer to this work.

is ubiquitous in nature, society and technology [20]. And the emergenceofsynchronizationpatternsinthesesystemshasbeen shown to be closely related to the underlying topology of in-teractions [21–23]. For example, community (a set of oscillators whichare placed atthe vertices) with highly densityof interac-tionssynchronizesmoreeasilythanthatwithsparseconnections, whichleadstotheapplicationoftheKuramotomodelindetecting communityandoverlappingincomplexnetworks[22–24].Inthis paper,we focusontheanalysisofthedifferenceofthelocal syn-chronization pacesandproposea dynamicalapproach toidentify theimportanceofverticesinacomplexnetwork.Thefundamental ideaofourdynamicalapproachisthatvertices’centralityis calcu-latedbasedontheDSA.Weanalyzetherobustnessandefficiency ofourdynamicalapproachthroughapplyingittoidentify commu-nity inthe PPIN, the DSN andthe benchmark network. We find that our approachhas powerfuladvantages oftimely, robustness andefficiency to identify vital vertices, forinstance the overlap-pingcommunity,incomplexnetworks.

2. Methodologyandanalysis

One of the simple paradigm to understand the synchroniza-tionphenomenonistheKuramotomodel[25,26],whichhasbeen appliedtostudythesynchronizationpatternsandtoidentify com-munity in complex networks. Consider a network of N vertices

joinedinpairsby

E links,

whichrepresenttheinteractionbetween vertices,forexample,theacquaintanceorcollaborations between individualinsocialnetworks.Ingraphdescription,thenetworkcan berepresentedbymeansofa

N

×

N connectivity matrixA, where

Ai j

=

1 whenvertices i and j are linked,and Ai j

=

0,otherwise.

Eachvertex, says i,isencodedaphaseoscillatorwiththenatural frequency

ω

iandphase

θ

i.Thephase

θ

ievolvesintimeaccording

totheKuramotomodel:

d

θ

i

(

t

)

dt

=

ω

i

+

λ

ki



j∈i sin

(θ

j

(

t

) − θ

i

(

t

))

(1)

where,



i is the subgraph of vertex i’s nearest neighbors, ki is

theconnectivityofvertex

i and

λ

isthepositivecouplingstrength betweenvertices. Notethatverticesare representedby their cor-responding oscillators in the following part of our paper. Many previous literatures show that there exists a critical coupling

λ

c,

above which synchronizationemerges spontaneously [25,26]. We hererealizetheKuramotomodel oncomplexnetworksusingthe Runge–Kutta methods. Without lack of generality, the coupling strength

λ

=

2

(> λ

c

)

isfixed,andthenaturalfrequencies

ω

is and

theinitial phase

θ

is are chosen froma uniform distribution g

(

x

)

withmean



x



=

0 intheinterval

(−

0

.

5

,

0

.

5

)

and

(−

π

,

π

)

, respec-tively.

Themacroscopiccomplexorderparameter,whichdescribesthe synchronizedbehaviorofthewholesystem,isdefinedas:

R

(

t

)

ei(t)

=

1 N N



j=1 eiθj(t) ₍₂₎

whereR

(

t

)

(0

≤

R

(

t

)

≤

1)quantifies theextentofsynchronization inasystemof N oscillators, and

(

t

)

istheaveragephaseofthe systemattime t.The largerthe R

(

t

)

is,themoreoscillatorstend tosynchronizetoacommonphase.Inthespecialcaseof

R

(

t

)

=

1, alloscillatorssharethesamephaseandthesystemreachesthe co-herencestate.However, R

(

t

)

doesnotgiveanyfurtherinformation aboutthesynchronizationbehaviorintermsoflocalcluster,which plays a crucial importance to understand the structural and dy-namicalroleofheterogeneityincomplexnetworks.Forthisreason,

Fig. 1. (Coloronline.) There-scaledDSAbetweenpairsofoscillatorsintheDSNat thedynamicaltimet=300.Thecolorsareagradationbetweencyan(0) andthe red (1).

insteadofconsideringtheglobalorderparameter,thereal-valued localorderparameter

r

j

(

t

)

attime

t is

definedasfollows:

rj

(

t

)

eij(t)

=

1 kj

+

1



l∈ j eiθl(t) ₍₃₎ where,





j is a subgraph including oscillator j and its nearest

neighbors,



j

(

t

)

describesthe averagephase ofoscillatorsin the

givensubgraph





j,and

r

j

(

t

)

describestheextentof

synchroniza-tion of the set of oscillators surrounding oscillator j at time t. Further,the localorderparameter

r

j

(

t

)

,whichrepresentsthe

dy-namicalfunction ofoscillator j, can beused toquantifythe syn-chronization abilityofoscillator j at time t. Takea networkwith a clearcommunitystructureforexample,oscillators belongingto thesamecommunitytendtosharethesamesynchronization abil-ity,andnotnecessarilyequalforallcommunities[23].Whilesome oscillatorsoftenbelong tomorethan onecommunityandcanbe calledtheoverlappingoscillators.Thoseoverlappingoscillatorswill haveaweaksynchronizationabilityduetothelimitedfrom differ-entcommunities.WeherefocusontheDSAamongoscillatorsand donotconsiderthesimilarcorrelationbetweenoscillators[22].

TheDSAbetweenoscillators,says

i and

j, iswrittenas,



ri j

(

t

) = |

ri

(

t

) −

rj

(

t

)|

(4)

Andweoperateasimplealgebraic calculationtore-scaleallthose



ri j

(

t

)

tofallintherangeof

[

0

,

1

]

,



_r

i j

(

t

) =

ri j

(

t

) −

rmin

(

t

)

rmax

(

t

) −

rmin

(

t

)

(5) where, rmin

(

t

)

=

Min

{

ri j

(

t

),

∀

i

,

j

}

and rmax

(

t

)

=

Max

{

ri j

(

t

),

∀

i

,

j

}

.

Take a complex network withclear communitystructure for ex-ample, it is obvious that oscillators, says i and j, in the same communityhavethesimilarsynchronizedabilitywiththesmaller



_r

i j

(

t

)

attime t.

InFig. 1,werepresent



ri j

(

t

)

at

t

=

300 fortheDSN[27].We

can identifythat the oscillatorswithNo. 22 and23have alarge gapaboutthe re-scaled DSAwithother oscillators, whichclearly indicates that the two oscillators have a large probability to be

the overlapping ones andplay the bridge role among communi-ties.Ourpreliminaryresultheredemonstratesthatthedynamical functionofthe re-scaledDSAplaysa positiveeffectto revealthe underlyingstructuralfunctiondirectly.

Then,wedefinethequantity

η

i

(

t

)

,whichdescribestheaverage

re-scaled DSAof oscillator i in the system, asthe below mathe-maticalform,

η

i

(

t

) =

1 N N



j=1



ri j

(

t

)

(6)

AnalogytotheDSA,wealsore-scaled

η

i

(

t

)

tofallintherange

of

[

0

,

1

]

,

η

 i

(

t

) =

η

i

(

t

) −

η

min

(

t

)

η

max

(

t

) −

η

min

(

t

)

(7)

where,

η

min

(

t

)

=

Min

{η

i

(

t

),

∀

i

}

and

η

max

(

t

)

=

Max

{η

i

(

t

),

∀

i

}

.

Andthen, wedefine thedynamical centralityofoscillator

i at

time

t as

follows



η

 i



t

=

1 t t



τ=1

η

 i

(

τ

)

(8)

whichistheaverageaccumulativevalueof

η

_i

(

t

)

andhasa poten-tialapplicationtorevealoscillator

i’s

topologicalstatusinasystem attime

t.

Inthespecialcase,oscillatorwiththeconnectivity

k

=

1 is a leaf andits

η





t is set to zero in the givennetwork. There

are two foldsof advantage of our presentapproach: one is that we buildthe simplecorrelation betweendynamics andstructure, whichshowsthatdifference(orheterogeneity)playsanimportant roleinrevealingtheinterplaybetweenthedynamicsandstructure ofcomplexsystem;theotheristhatthedefinitionof

η

_i



t allows

ustotracethetimeevolutionofeachoscillator’sdynamicalstatus andtoidentifyvertices’topologicalstatusfromthedynamical per-spectivetimely.Ourdynamicalapproach providesapractical way tounveilthepotentialinteractionsbetweenverticesthroughtheir dynamicalbehaviors,suchasthebehaviorofsendingandreceiving lettersin theemail networksandtheprocess ofthe biochemical reactionintheprotein–proteininteractionnetwork.

InFig. 2,we represent

η

_i



t of eachoscillator inthe four

dif-ferentcomplexnetworks:theDSNwiththesize N

=

62,thePPIN withN

=

163,thenetworkofcoauthorshipsbetween

N

=

379 sci-entists whose research centers on the top of networks [28], the benchmark network with N

=

128, the college football network with N

=

115 [29] and the Barabasi–Albert scale-free network (BASFN)with N

=

100,attime

t

=

300. The benchmarknetwork is built with N

=

128 vertices divided into four communities of 32 verticeseach.Verticesinthesamecommunityarelinkedwith probability pin and vertices in different communities are linked

withprobability pout [29].AndtheBASFNisconstructedbasedon

two basic principles of growth andpreferential attachment [30]. The value of each oscillator’s

η

_i



t is proportional to the radius

ofthecorrespondingcircle. Thedifferenceamongoscillators’

η

_i



t

in BASFN issmaller than that inthe other fivenetworks witha clearcommunitystructure.Whiletheoscillatorswithhighervalue of

η

_i



t mostlylie onthemarginalofcommunityincomplex

net-workwithaclearcommunitystructure,suchastheDSN,thePPIN

[31],thenetworkofcoauthorshipsbetweenscientists[28]andthe benchmarknetworkwithparameters pin

=

0

.

05 and pout

=

0

.

005.

Thoseoscillatorswithhighervalueof

η

_i



t calledtheoverlapping

oscillatorsplaythebridgeroleinconnectingmotifs.

Beforeanalyzingtheapplicationofourdynamicalapproach,we shouldstudytherobustness ofourapproachduetothepresence of initial dynamical noise andthe stochastic effects. We realized theKuramoto model50 times independentlyinthe DSNandthe

PPIN,respectively.Foreachdynamicaltime

t,

wesortallthe oscil-lators frombigtosmallaccordingtotheir

η

_i



t.Wethenanalyze

the probabilities that each oscillatorfalls inthe top ten P10 and

in thetoptwenty P20evolve asthedynamical time

t elapses.

In

Fig. 3, we find that P10 and P20 of those given oscillators, such

astheoscillator120 inthePPINandtheoscillator21 intheDSN, reach their own steady values when t

>

100 respectively, which certifiestherobustnessofourdynamicalapproachtoquantify ver-tices’centrality.

Meanwhile,inordertotestourapproach’saccuracyinrevealing topological structure, we calculate the betweenness centrality of each vertex,says Bc

(

v

)

forvertex v, inanundirectednetwork as

thefollowingformula[4],

Bc

(

v

) =

2

(

N

−

1

)(

N

−

2

)



s =v =d nsd

(

v

)

nsd (9) where nsd isthe totalnumber ofshortestpaths fromvertexs to

vertex

d, n

sd

(

v

)

isthenumberofthosepathspassthrough v.We

calculatedthebetweennesscentralityBc andtheprobabilities P10

and P20 for each vertex respectively. For some known vital

ver-tices (Table 1), forexample,vertex23 in theDSN andvertex64 in the PPIN,we found that verticeswithlarge betweenness cen-tralityhavelarge P10 and P20,i.e., thosevital verticeshavelarge

probability of being identified by means of our approach. It is surprising that there also exist some exception vertices (suchas vertex 21 and vertex 24 in the DSN) have large dynamical cen-tralitybuthavesmallbetweennesscentralityduetothedifference between our approach and the betweenness centrality index. In the betweenness centrality index, the betweenness centrality is calculated based on the globalinformation aboutthe topological structure whichishiddenina blackbox.While inourdynamical approach,thedynamicalcentralityiscalculatedjustonlybasedon vertices’statusandtheir localdynamicalabilities. Althoughthere exists a clear difference betweenthe definitions ofour approach andthebetweennesscentralityindex,ourapproachhasa satisfac-toryperformanceto identifyvitalverticalwithlargebetweenness centrality,whichshowsthenoveltyofourapproachtoreveal ver-tices’topologicalpropertyincomplexnetworks.

3. Application

Due to our approach’s satisfactory performance in identifying overlapping vertices, we shedlight on theapplication ofour ap-proachtodetectcommunityincomplexnetworks,wherevertices are dividedintogroups. Verticesaremoretightlyconnectedwith each other in the same group, meanwhile there also exist some overlappingverticesbelongingtomorethanonegroup.Further,in ordertoquantifythevalidity ofpossiblesubdivisions, weusethe conceptofmodularity

Q as

afunctionform[32,33]

Q

=



i

(

eii

−

a2i

)

(10)

where

a

i

=



jei j isthe totalfractionoflinkswithonevertexin

community i in network and ei j is the fraction of all linksthat

linkverticesincommunity

i to

verticesincommunity j.The mod-ularity Q is a practical index to assessa givendivision intoany number of communities for a given network. Larger value of Q

indicatesstrongercommunitystructure[34].

Combining with the process ofquantifying vertices’centrality and the process of community detection, we introduce two dif-ferent times: the dynamical time t and the detecting action n. The algorithm ofidentifying communityusingthe dynamical ap-proach toquantify vertices’centralityis simplystatedasfollows. Asmentionedabove,eachvertexdenotedby

i is

encodedaphase oscillator withits naturalfrequency

ω

i andphase

θ

i.Step 1. The

Fig. 2. (Coloronline.) Anintuitivedisplayofη

itofeachoscillatorin(a)theDSN,(b)thePPIN,(c)thenetworkofcoauthorshipsbetweenscientistswhoseresearchcenters

onthetopofnetworks,(d)thebenchmarknetwork,(e)thecollegefootballnetworkand(f)theBAscale-freenetworkatthedynamicaltimet=300.Thenumberlabeledon eachnodeisthenumericalcodeofthecorrespondingnodein(a)and(b)andthevalueofeachoscillator’sη

it(forall i)isproportionaltotheradiusofthecorresponding

circle.

dynamicalprocess ofthe Kuramotomodel.Eachoscillator, says

i,

updatesits phase

θ

i

(

t

)

according to Eq. (1), and then we

calcu-late

η





ts forall oscillators atthe dynamical time t. Step 2. The

detecting process of community. All the oscillators are sorted as adecreasingorderaccordingto their

η





ts. Andthen,all the

os-cillatorsare operatedinturnto identifycommunity:forthe

n-th

oscillator,we keepthe onlyonelink thathasthe largestnumber ofcommonneighborsanddeletealltheotherlinks,andcalculate the modularity Qt

(

n

)

at the detecting action n, till all the

oscil-lators areoperated.Step 3.All thedeletedlinksare repairedand

Fig. 3. (Coloronline.) Theprobability(P10)thateachgivenoscillatorfallsinthetoptenevolvesasdynamicaltimet elapsesin(a)thePPINand(c)theDSN.Andthe probability(P20)thateachgivenoscillatorfallsinthetoptwentyevolvesasdynamicaltimet elapsesin(b)thePPINand(d)theDSN.Thecodesoftheseoscillatorsarethe sameasthatinFig. 2(a)andFig. 2(b),respectively.Redlinesareguidestotheeye.

Table 1

Thistableshowsthe probabilitiesthatsome knownvitaloscillators(codes:No.) fallinthetopten(P10)andinthetoptwenty(P20)inthePPINandintheDSN atthedynamicaltimet=300.Bc isthescoreofbetweennesscentralityforthe

correspondingvertex.Thecodesoftheseoscillators(No.)arethesameasthatin Fig. 2(a)andFig. 2(b),respectively.

PPIN DSN No. P10 P20 Bc No. P10 P20 Bc 64 98% 100% 0.565 24 66% 74% 0.076 105 94% 96% 0.510 23 64% 76% 0.250 100 78% 94% 0.497 18 62% 70% 0.209 120 74% 88% 0.394 20 54% 62% 0.122 129 70% 80% 0.333 21 50% 66% 0.034 143 54% 76% 0.229 27 36% 56% 0.146

thebothprocessesarerepeatedfromstep1asthedynamicaltime

t elapses. Notethat each link isoperated one time whetherit is deletedornotandall themutualinteractions betweenoscillators arenotdestroyedduringthedetectingprocessofcommunity.

Weanalyzetheevolutionofthemodularity Qt

(

n

)

asafunction

ofthedetecting action

n at

eachdynamicaltime

t in

thePPIN.In

Fig. 4, we plotthe relationshipbetween Qt

(

n

)

and thedetecting

action

n at

thedynamicaltime

t

=

10

,

20

,

50 and100,respectively. Wefindthat Qt

(

n

)

reachesitsmaximumvalueatabout

n

=

10 in

the PPIN. Namely, there exists a peak with the maximum value of Qt

(

n

)

when we operate the community detecting process on

thetoptenoscillators,whichshowsthatourdynamicalapproach has its better performance in identifying the overlapping oscilla-tors. Another surprising result is that the modularity Q reaches

its maximum value at t

=

20 during the dynamical process of quantifying vertices’ centrality, which shows that our dynamical approach has highefficiency although there exists some internal initial noise. Notethat our results reveal thepositive role of dy-namicsprocessinreconstructingthetopologicalstructureofthose interaction units in complex networks. Since structural informa-tion isoftenhidden inthe dynamicalbehavior ofthe vertices in an unknown manner [16,35,36], we recognize that vertices with

Fig. 4. (Coloronline.) ThemodularityQ evolvesasafunctionofthedetectingaction n intheprotein–proteininteractionnetworkwhenoscillatorsevolveatthe dynam-icaltimet=10,20,50 and100,respectively.Theredlineshowsthatthevalueof themodularityQ whenweoperatethetoptenoscillatorsusingthedetecting pro-cess.

thesimilardynamicalpropertieshavealargeprobabilitytohavea similartopologicalstructure.

4. Conclusion

In summary, we proposed a dynamical approach to identify vertices’ centralityduring the synchronizationprocess of the Ku-ramoto model.Differentfromsome previous literatures, the nov-eltyofourdynamicalapproachisthatvertex’scentralityis quanti-fiedbasedontheDSA.Weanalyzedtherobustnessandefficiency ofourdynamical approachthroughapplyingit todetect commu-nityinthePPINandtheDSN.Furthermore,wefoundthatour ap-proachhasabetterperformancetoidentifytheweaklinks,which endwithverticeswithahigherdynamical centrality.Namely,the weak linksconnecttheoverlapping verticesincomplexnetworks witha clearly communitystructure.Ourpresentworkprovides a newperspectiveandtoolstounderstandthebridgingroleof het-erogeneity (i.e., difference) in revealing the connection between the dynamical behavior andthe underlying structure of elemen-taryunitsincomplexnetworks.

Acknowledgements

L. Guo and W.Y. Zhang wish to acknowledge the financial support by the China Scholarship Council under Grant Nos. 201606415063and201406340126.

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