Effect of correlations on network controllability.

1.3 MB·English

by  Posfai, Marton

Effect of correlations on network
controllability
SUBJECTAREAS:
Ma´rtonPo´sfai1,2,3,7,Yang-YuLiu1,4,Jean-JacquesSlotine5&Albert-La´szlo´ Baraba´si1,6,7
STATISTICALPHYSICS,
THERMODYNAMICSAND
NONLINEARDYNAMICS
1CenterforComplexNetworkResearchandDepartmentofPhysics,NortheasternUniversity,Boston,2DepartmentofPhysicsof
APPLIEDPHYSICS ComplexSystems,Eo¨tvo¨sUniversity,Budapest,3DepartmentofTheoreticalPhysics,BudapestUniversityofTechnologyand
Economics,Budapest,4CenterforCancerSystemsBiology,Dana-FarberCancerInstitute,Boston,5Non-linearSystemsLaboratory,
OTHERPHYSICS
DepartmentofMechanicalEngineeringandDepartmentofBrainandCognitiveSciences,MassachusettsInstituteofTechnology,
PHYSICS Cambridge,6DepartmentofMedicine,BrighamandWomen’sHospital,HarvardMedicalSchool,Boston,7CenterforNetwork
Science,CentralEuropeanUniversity,Budapest.
Received
24August2012 Adynamicalsystemiscontrollableifbyimposingappropriateexternalsignalsonasubsetofitsnodes,itcan
bedrivenfromanyinitialstatetoanydesiredstateinfinitetime.Herewestudytheimpactofvarious
Accepted networkcharacteristicsontheminimalnumberofdrivernodesrequiredtocontrolanetwork.Wefindthat
10December2012 clusteringandmodularityhavenodiscernibleimpact,butthesymmetriesoftheunderlyingmatching
problemcanproducelinear,quadraticornodependenceondegreecorrelationcoefficients,dependingon
Published
thenatureoftheunderlyingcorrelations.Theresultsaresupportedbynumericalsimulationsandhelp
15January2013
narrowtheobservedgapbetweenthepredictedandtheobservednumberofdrivernodesinrealnetworks.
Correspondenceand While during the past decade significant efforts have been devoted to understanding the structure,
requestsformaterials evolution and dynamics of complex networks1–6, only recently has attention turned to an equally
shouldbeaddressedto important problem: ourability to control them. Given theproblem’s importance, recent work has
A.-L.B.([e-mail]) extendedtheconceptofpinningcontrol7–9andstructuralcontrollability10–13tocomplexnetworks.Herewefocus
onthelatterapproach.Anetworkedsystemisconsideredcontrollableifbyimposingappropriateexternalsignals
onasubsetofitscomponents,calleddrivernodes,thesystemcanbedrivenfromanyinitialstatetoanyfinalstate
infinitetime14–17.Asthecontrolofasystemrequiresaquantitativedescriptionofthegoverningdynamicalrules,
progressinthisareawaslimitedtosmallengineeredsystems.Yet,recentlyLiuetal.10showedthattheidentifica-
tionoftheminimalnumberofdrivernodesrequiredtocontrolanetwork,N ,canbederivedfromthenetwork
D
topologybymappingcontrollability16tothemaximummatchingindirectednetworks18.Themappingindicated
thatN ismainlydeterminedbythedegreedistributionP(k ,k ).Weknow,however,thataseriesofchar-
D in out
acteristics,fromdegreecorrelations19–21tolocalclustering22andcommunities23–26,cannotbeaccountedforby
P(k ,k )alone,promptingustoask:whichnetworkcharacteristicsaffectthesystem’scontrollability?
in out
The three most commonly studied deviations from the random network configuration are (i) clustering,
manifestedasahigherclusteringcoefficientCthanexpectedbasedonthedegreedistribution27;(ii)community
structure, representing the agglomeration of nodes into distinct communities, captured by the modularity
parameterQ25;(iii)degreecorrelations28.First,wemotivateourworkbyshowingthatnetworkcharacteristics
otherthanthedegreedistributionalsoaffectnetworkcontrol.Next,weusenumericalsimulationstoidentifythe
networkcharacteristicsthataffectcontrollability,findingthatonlydegreecorrelationshaveadiscernibleeffect.
Wethenanalyticallyderiven 5N /Nforrandomnetworkswithagivendegreedistributionandcorrelation
D D
profile.MoredetailedcalculationsareprovidedintheSupplementaryInformationSec.III.Finally,wetestour
predictionsonrealnetworks.
Results
Prediction based on the degree distribution. To motivate our study we compared the observed N to the
D
predictionbasedonthedegreesequenceforseveralrealnetworks.Forthiswerandomizeeachnetworkpreserving
itsdegreesequenceandwecalculateNrand,thenumberofdrivernodesfortherandomizednetwork.PlottingN
D D
versusNrandonlog-logscaleindicatesthatthedegreesequencecorrectlypredictstheorderofmagnitudeofN
despitekDnowncorrelations19,20(Fig.1a).However,byplottingn 5N /Nversusnrand~Nrand(cid:2)N weobservDe
D D D D
clear deviations from the degree based prediction (Fig. 1b). Our goal is to understand the origin of these
deviations,andthedegreetowhichnetworkcorrelationscanexplaintheobservedn .
D
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Figure1|(a)WecompareN forrealsystemstoNrand,representingthenumberofdrivernodesneededtocontroltheirrandomizedcounterparts.
D D
Randomizationeliminatesalllocalandglobalcorrelations,onlypreservingthedegreesequenceoftheoriginalsystem.Wefindthatthedegreesequence
predictstheorderofmagnitudeofN correctly,however,smalldeviationsarehiddenbythelogscale,neededtoshowthewholespanofN seeninreal
D D
systems.(b)Thesedeviationsaremoreobviousifwecomparethedensityofdrivernodesn 5N /Nandnrandinlinearscale,findingthatforsome
D D D
systems(e.g.regulatoryandp2pInternetnetworks)thedegreesequenceservesasagoodpredictorofn ,whileforothersystems(e.g.metabolicnetworks
D
andfoodwebs)n deviatesfromthepredictionbasedsolelyonthedegreesequence.
D
Numerical simulations. We start from a directed network with whereA istheadjacencymatrix,c andc arethecommunitiesthe
vw v w
Poisson29,30 or scale-free degree distribution31,32. The scale-free vandwnodesbelongto,respectively.SpecifyingQstillleavesagreat
networkisgeneratedbythestaticmodeldescribedintheMethods amountoffreedominthenumberandsizeofthecommunities.We
section. We use simulated annealing to add various network thereforechoosetorandomlydividethenodesintoN equallysized
C
characteristics by link rewiring, while leaving the in- and out- groups,andincreasetheedgedensitywithinthesegroups,elevating
degrees unchanged, tuning each measure to a desired value, for Qtothedesiredvalue.
details see the Methods section. We computed n using the The simulations indicate that this community structure has no
D
Hopcroft-Karpalgorithm33. effectonn (Fig.2b).Whileaddingcommunitiestonetworkscan
D
be achieved in many different ways, and the effect of modularity
Clustering. We use the global clustering coefficient27 defined for can be explored in more detail (e.g. hierarchical organization of
directednetworksas communities23,34,35, overlapping community structure24,36, etc), we
C~ 3:numberof triangles : ð1Þ have failed to detect systematic, modularity induced changes in
2:numberof adjacentedgepairs n ,prompting us to conclude that Qdoes not playa leading role
D
in n .
D
ThesimulationsindicatethatchangesinConlyslightlyaltern and
D
thattheeffectisnotsystematic(Fig.2a).HenceweconcludethatC
playsanegligibleroleindeterminingn . Degreecorrelations.Indirectednetworkseachnodehasanin-degree
D
(k) and an out-degree (k ), thus we can define four correlation
i o
Modularity.Wequantifythecommunitystructureusing25,26: coefficients: correlations between the source node’s in- and out-
" # degree, and the target node’s in- and out-degree (Figs. 3, 4)28. We
Q~1X A {kðvinÞkðwoutÞ d , ð2Þ usethePearsoncoefficienttoquantifyeachcorrelationwithasingle
E vw vw E cv,cw parameter:
Figure2|EffectoftheclusteringcoefficientCandmodularityQonthedensityofdrivernodes,n . NetworksizeisN510,000.Eachdatapointisan
D
averageover50independentruns;theerrorbars,typicallysmallerthanthesymbolsize,representthestandarddeviationofthemeasurements.
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Figure3|Theimpactofdegree-degreecorrelationsonthedensityofdrivernodes(n )fortheErdo˝s-Re´nyimodel(N510,000)foraveragedegreesÆkæ
D
51(red),Ækæ53(green),Ækæ55(blue),Ækæ57(black)andÆkæ59(orange).Theresultsaresimilarforthescale-freemodel(seeFig.4).Eachdatapoint
isanaverageof100independentruns.
1X (cid:3) (cid:4)(cid:3) (cid:4) understanding analytically the role of degree correlations, which,
kðaÞ{kðaÞ jðbÞ{jðbÞ
rða{bÞ~E e e e , ð3Þ bysystematicallyalteringnD,affectthesystem’scontrollability.
sðaÞsðbÞ
X
where ?sumsoveralledges,a,bg{in,out}isthedegreetype, Analyticalframework.Thetaskofidentifyingthedrivernodescan
e
k(a) is the degree of thesource node, j(b) is the degree ofthe target bemappedtotheproblemoffindingamaximummatchingofthe
1X network10.Amatchingisasubsetoflinksthatdonotsharestartor
node. And ja~E ejae is the average degree of the nodes at the endpoints.Wecallanodematchedifalinkinthematchingpoints
1X (cid:3) (cid:4)2 at it and we gain full control over a network if we control the
beginningofeachlink,s2~ kðaÞ{kðaÞ isthevariance;kðbÞ
a E e e unmatched nodes. The cavity method has been successfully used
ands(b)aredefinedsimilarly. to calculate the size of the maximum matching for undirected37
SimulationsshowninFigs.3and4indicatethatdegreecorrela- and directed10 network ensembles with given degree distribution.
tions systematically affect n . We observe three distinct types of Here we study network ensembles with a given degree correlation
D
behavior: profile.
Wecalculaten analyticallyforagivenP(k ,k )andselected
(i) n depends monotonically on r(out-in), so that low (negative) D in out
D degree-degreecorrelatione(j ,j ;k ,k ),representingtheprob-
correlationsincreasen andhigh(positive)correlationslower in out in out
D abilityofadirectedlinkpointingfromanodewithdegreesj andj
n (Figs.3c,4c); in out
D toanodewithdegreesk andk .Intheabsenceofdegreecorrela-
(ii) Bothr(in-in)andr(out-out)increasen ,independentofthesignof in out
D tions(neutralcase)
thecorrelations(Figs.3a,3d,4a,4d);
(iii) r(in-out)hasnoeffectonnD(Figs.3c,4c). eð0Þðji,jo;ki,koÞ~PðinÞðjiÞQðoutÞðjoÞQðinÞðkiÞPðoutÞðkoÞ, ð4Þ
ThebehaviorisqualitativelythesameforErdo˝s-Re´nyi(Fig.3)and
scale-free(Fig.4)networks. where QðoutÞðj Þ~2joPðoutÞðj Þ, QðinÞðkÞ~2kiPðinÞðkÞ and Ækæ is
Thediversityofthesenumericalresultsrequireadeeperexplana- o hki o i hki i
tion. Therefore in the remaining of the paper we focus on theaveragedegree.Toensureanalyticaltractabilitywechose21
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Figure4|Theimpactofdegree-degreecorrelationsonthedensityofdrivernodes(n )forthescale-freemodel(N510,000,c52.5)foraverage
D
degreesÆkæ51(red),Ækæ53(green),Ækæ55(blue),Ækæ57(black)andÆkæ59(orange). TheresultsaresimilarfortheErdo˝s-Re´nyimodel(seeFig.3).
Eachdatapointisanaverageof100independentruns.
eðin-inÞðj,j ;k,k Þ~QðoutÞðj ÞPðoutÞðk Þ
i o i o o o
?
hPðinÞðjÞQðinÞðkÞzrðin-inÞmðin-inÞðj,kÞi, ð5aÞ sðaÞsðbÞ Xjk:mða{bÞðj,kÞ~1, ð7Þ
i i i i
j,k~0
eðin-outÞðj,j ;k,k Þ~QðoutÞðj ÞQðinÞðkÞ andallelementsofe(a–b)(j,k)arebetween0and1.
i o i o o i Ourgoalistounderstandtherelationbetweenn andthedegree
hPðinÞðjÞPðoutÞðk Þzrðin-outÞmðin-outÞðj,k Þið,5bÞ correlationcoefficientr(a–b).Assumingthatr(a–b)issDmallwetreatthe
i o i o
correlations as perturbations to the neutral case, discussing the
impactofthefourr(a–b)correlationsseparately.
eðout-inÞðj,j ;k,k Þ~PðinÞðjÞPðoutÞðk Þ
i o i o i o
h ið5cÞ Out-incorrelations.Usingequation(5c)andkeepingthefirstnon-
QðoutÞðj ÞQðinÞðkÞzrðout-inÞmðout-inÞðj ,kÞ , zerocorrectionweobtain(SupplementaryInformationSec.III.):
o i o i
hki
n ðout(cid:2)inÞ~n ð0Þ{rðout(cid:2)inÞ ½M ðw^ ,1{w ÞzM ð1{w^ ,w Þ(cid:3),ð8Þ
D D 4 1 2 1 1 1 2
eðout-outÞðj,j ;k,k Þ~QðinÞðjÞPðinÞðkÞ
i o i o i i
hPðoutÞðjoÞQðoutÞðkoÞzrðout-outÞmðout-outÞðjo,koÞið:5dÞ wwhorekre;wnDað0nÞdisw^thoenflryacdteipoennodfodnriPve(krn,okdes)o10f,tahneduncorrelatednet-
i i in out
?
Byfixingm(a–b)(j,k)(a,bg{in,out})weobtainaoneparameter M ðx,yÞ~Xmðout(cid:2)inÞðj,kÞxj{1yk{1: ð9Þ
1
networkensemblecharacterizedbyr(a–b),wherem(a–b)(j,k)satisfies
j,k~1
theconstraints
X? mða{bÞðj,kÞ~X? mða{bÞðj,kÞ~0, ð6Þ Esuqpupaotirotend(8b)yprseimdiuctlasttihoantsnfDodrespmenaldlsrl(ionute-ian)rl(yFoignsr.(o3uct-ina),nadp4recd).icTtihoins
j~0 k~0 behavior is also revealed by the equivalent problem of finding the
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two-step correlations, accounting for the symmetry of the effect
observedinsimulations(Figs.3dand4d).
In-in correlations. Switching the direction of each link does not
changethematching,butturnsout-outcorrelationsintoin-incor-
relations. So n in(cid:2)in can be obtained by exchanging P(in)(k ) and
D in
P(out)(k )inequation(10),predictingagainaquadraticdependence
out
Figure5|One-stepout-outcorrelationsinducepositivetwo-step onr(in-in),supportedbythenumericalsimulations(Figs.3aand4a).
correlation. Positive(negative)correlationbetweenneighboringnodes
meansthatifnodeAhashighout-degree,thennodeBislikelytohavehigh In-outcorrelations.Theequationsforn donotdependonthein-
D
(low)out-degree,andhenceCwilllikelyhavehighout-degree. degreeofthesourceandtheout-degreeofthetargetofalink,hence
wepredictthatr(in-out)doesnotplayaroleinnetworkcontrollability,a
predictionsupportedbythesimulations(seeFigs.3band4b).
maximummatchingofgraphs10.ForanodeAwithout-degreek ,by
0 Takentogether,wepredictthatthefunctionaldependenceofn
definitiononlyoneedgecanbeinthematching.Iftheremainderk – D
0 ondegreecorrelationsdefinesthreeclassesofbehaviors,depending
1edgespointtonodeswithdegree1(disassortativecase),Ainhibits
on the matching problem’s underlying symmetries: n has no
themfrombeingmatched,sowehavetocontroleachofthemindi- D
dependence on r(in-out), linear dependence on r(out-in) and quadratic
vidually,increasingn .Iftheremainderk –1edgespointtohubs
D 0 dependence on r(in-in) and r(out-out). These predictions are fully sup-
(assortative case), these hubs are likely to be matched through
portedbynumericalsimulations(Figs.3and4):forsmallrweseeno
anotherincomingedge,decreasingn .
D dependence on r(in-out), an asymmetric, monotonic dependence on
Out-out correlations. The cavity method indicates that for out-out r(out-in),andasymmetriconr(in-in)andr(out-out).
correlationsthefirstnonzerocorrectionisoforder(r(out-out))2: Todirectlycomparetheanalyticalpredictionstosimulationswe
needtoknowthecompletee(j,j ;k,k )distribution,whichisnot
i o i o
n ðout-outÞ~n ð0Þzrðout-outÞ2hki explicitlysetinoursimulations.Sototesttheresultsweusearewir-
D D 8 ing method that sets the e(j, j ; k, k ) distribution, not only the r
ð10Þ i o i o
h i correlationcoefficient21.Thismethodisnotasrobustasouroriginal
HðinÞ’ð1{w ÞM ðw^ ÞzHðoutÞ’ðw ÞM ð1{w^ Þ ,
1 2 2 2 2 1 algorithm and the range of accessible r values is more restricted.
However,sinceourresultsarebasedonperturbationschemeweonly
whereHðaÞðxÞ~P?k~1QðaÞðkÞxk{1(ag{in,out})onlydependson expectthemtobecorrectforsmallrvalues.Indeed,wefindthatthe
P(kin,kout)and predictionsquantitativelyreproducethenumericalresultsinafair
M ðxÞ~ X? mðout(cid:2)outÞðl,jÞmðout(cid:2)outÞðl,kÞxj{1xk{1: ð11Þ intervalofr(a–b)(Fig.6).
2 PðoutÞðlÞ
j,k~1,l~0 Realnetworks.Wetestthepredictionsprovidedbythedeveloped
analyticalandnumericaltoolsonasetofpubliclyavailablenetwork
Equation(10)predictsthatnDðout(cid:2)outÞ doesnotdependontheout- datasets.Whencomplexsystemsaremappedtonetworks,thelinks
out correlation of the directly connected nodes, but only on the connecting thenodes represent interactions between them. In this
correlationbetweenthesecondneighbors,henceitsdependenceis context self-loops represent self-interactions, with a strong, well
quadratic in r(out-out), a prediction supported by numerical simula- understood impact on controllability10,38. While in some systems
tions(Figs.3dand4d).Indeed,positive(negative)r(out-out)correlation self-loops are obviously present (e.g. neural networks), in others
betweentheimmediateneighborsmeansthatifnodeAhashighout- they are manifestly absent (e.g. electric circuits39). Our purpose
degree,thennodeBisexpectedtohavehigh(low)out-degree,and here is to test the effect of correlations, hence we rely on datasets
thereforeCislikelytohavehighout-degree(Fig.5).Thatis,both that capture the wiring diagram of various complex systems with
positiveandnegativeone-stepout-outcorrelationsinducepositive differentcorrelationproperties.Therefore,evenifinafewofthese
Figure6|Theanalyticformulasaretestedwithsimulationsonan(a)Erdo˝s-Re´nyimodelandona(b)scale-freemodel. Weusedthealgorithm
proposedin21tosete(a2b)(j,j ;k,k ).For(a)networkwechooseN51,000andÆkæ53;for(b)N51,000,c52.5andÆkæ54.Eachdatapointisan
i o i o
averageover100independentruns;theerrorsrepresentbythestandarddeviationofthemeasurements.
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Figure7|TheobservedandpredicteddeviationbetweenN andNrand. Redline:D~(cid:5)N {Nrand(cid:6)(cid:2)N,thepredictionerrorbasedonthedegree
D D D D
sequence.Dashedlines:correlationsrelevanttocontrollability.ForeachnetworkDiscalculatedbyaveragingover50independentconfigurations.
mapsself-loopsaremissing,itisbeyondthescopeofthisworkto correlations,henceweexpectn toincrease(D.0),consistentwith
D
completethesenetworks.However,whenstudyingcontrollabilityof theobservations.
aparticularsystem,carefulthoughthastobeputintowhetherself-
GroupE.Onlythetranscriptionalregulatorynetworksaresomewhat
loopsarepresentornot.Wepresentasystematicstudyontheeffect
puzzling in that they show degree correlations, yet the degree
ofself-loopsintheSupplementaryInformationSec.II.B.
sequencestillcorrectlygivesn .However,thesimulationsindicated
Totesttheimpactofourpredictionsonrealnetworkswecalculate D
that the effect of correlations is negligible for high n . And our
N {Nrand D
D~ D D , ð12Þ analytical results showed that the value of the correction depends
N ondetailsofe(j,j ;k,k ),notcapturedbythePearsoncoefficientr.
i o i o
These observations highlight that even though in most cases our
where Nrand represent the number of driver nodes for the degree-
D qualitative predictions based on r are valid, in some cases further
preservedrandomizedversionoftheoriginalnetwork.HenceifD5
investigationisrequired.
0 then P(k , k ) accurately determines N ; if D ? 0 then the
in out D
structuralpropertiesnotcapturedbythedegreesequenceinfluence
Discussion
itscontrollability.Wemeasure thecorrelationsinseveral realnet-
Thegoalofourpaperwastoclarifythehigherordernetworkchar-
worksandbasedonournumericalandanalyticalresultswepredict
acteristics that influence controllability. We studied the effect of
the sign of D (Fig. 7). We grouped the networks according to our
three topological characteristics: clustering, modularity and degree
predictions. Weprovide the details ofeach network dataset in the
correlations.Weusednumericalsimulationstoidentifytheroleof
SupplementaryInformationTableSI.
the relevant characteristics, finding that changes in the clustering
GroupA.Thenetworksofp2pInternet(Gnutellafilesharingclients) coefficient and the community structure have no systematic effect
donothave strongcorrelations, therefore weexpect nDtobe cor- onthetheminimumnumberofdrivernodesnD.Incontrastdegree
rectlyapproximatedbythepredictionbasedonP(k ,k )(i.e.D< correlationsshowedarobusteffect,whosemagnitudeanddirection
in out
0),inlinewiththeempiricalobservations. depends on the type of correlation. Using the cavity method we
derivedn fornetworkswithgivendegreedistributionandcorrela-
D
GroupB.Asinmostnetworksthethreerelevantcorrelationscoexist tionprofiles,findingresultsthatareconsistentwithournumerical
tosomedegree(Fig.7),itisimpossibletoisolatetheirindividualrole. simulations.Forrealnetworksthesenumericalandanalyticresults
Yet,thenetworksinthisgroup(electriccircuits,metabolicnetworks, enabledustoqualitativelyexplainthedeviationoftheobservedn
D
neuralnetworks,powergridsandfoodwebswithexceptionofthe fromthepredictionbasedonlyonP(k ,k ).
in out
Seagrass network) all have negative out-in and nonzero in-in and Ourresultsnotonlyofferanewperspectiveontheroleoftopo-
out-out correlations, each of which individually increase n as we logical properties on network controllability, but also raise several
D
showedabove.ThereforewepredictD.0,inlinewiththeempirical questions. Future research directions include determining the
observations. optimal network structure to minimize the number of necessary
driver nodes, and studying how different network characteristics
Group C. Only the prison social-trust and the cell phone network influencetherobustnessofthecontrolconfiguration.
featuresignificantpositiveout-incorrelations.Thesenetworksalso
displaynonzeroin-inandout-outcorrelation,leadingtothecoex- Methods
istenceoftwocompetingeffects:out-incorrelationsdecreasen and Generatingascale-freenetwork.Weusethestaticmodeltogeneratedirectedscale-
D freenetworks40.WestartfromNdisconnectednodesandassignaweightw 5(i1
theout-outandin-incorrelationsincreasen .Sincetheout-incor- i
D i)–atoeachnodei(i51…N).Werandomlyselecttwonodesiandjwithprobability
relationisafirstordereffect(equation(8)),whileout-outandin-in p0roportionaltow andw respectivelyandiftheyareyetnotconnected,weconnect
i j
correlations are only of second order (equation (10)), we expect a them.Weallowself-loops,butavoidmulti-edges.WerepeattheprocessuntilLlinks
decreaseinn (i.e.D,0),consistentwiththeempiricalresults. havebeenplaced.TheresultingnetworkhasaveragedegreeÆkæ52L/N,andP(in/out)(k)
D ,k–cforlargek,wherec~1za1,andmaximumdegreekmax*i{0a.
GroupD.TheSeagrassfoodwebandcitationnetworksdonotfeature
Tosystematicallystudycorrelations,thestartingnetworkhastobeuncorrelated.
significantout-incorrelations,onlythesecondaryin-inandout-out However,thepresenceofhubsmayinduceunwanteddegreecorrelations41,andmay
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