Heterogeneouspair-approximationforthecontactprocessoncomplexnetworksAngelicaS.Mata1,RonanS.Ferreira2andSilvioC.Ferreira1E-mail: angelica.mata@ufv.br,ronan.ferreira@ua.pt,silviojr@ufv.br1DepartamentodeFsica,UniversidadeFederaldeVicosa,36570-000,Vicosa,MG,Brazil2DepartmentofPhysics&I3N,UniversityofAveiro,3810-193Aveiro,PortugalAbstract. Recentworkshaveshownthatthecontactprocessrunningonthetopofhighlyheterogeneousrandomnetworksisdescribedbytheheterogeneousmean-eldtheory. However, some important aspects as the transition point and strong correctionstothenite-sizescalingobservedinsimulationsarenotquantitativelyreproducedinthistheory. Wedevelopaheterogeneouspair-approximation, thesimplestmean-eldapproachthattakesintoaccountdynamicalcorrelations,forthecontactprocess. Thetransition points obtained in this theory are in very good agreement with simulations.The proximity with a simple homogeneous pair-approximation is elicited showing thatthe transitionpoint insuccessive homogeneous cluster approximations moves awayfromthesimulationresults. Weshowthatthecriticalexponentsoftheheterogeneouspair-approximationintheinnite-sizelimitarethesameasthoseof theone-vertextheory. However, excellent matches withsimulations, for awide range of networksizes, areobtainedwhensub-leadingnite-sizecorrectionsgivenbythenewtheoryareexplicitlytakenintoaccount. Thepresentapproachcanbesuitedtodynamicalprocessesonnetworksingeneralprovidingaprotablestrategytoanalyticallyassessne-tuningtheoreticalcorrections.PACSnumbers: 89.75.Hc,05.70.Jk,05.10.Gg,64.60.an1. IntroductionTheaccuratetheoretical understandingof dynamical systemsintheformof reaction-diusionprocesses runningonthetopof complexnetworks rates amongthehottestissues in complex network theory [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Much eort hasbeendevotedtothecriticalityoftheensuingabsorbingstatephasetransitionobservedin the contact process (CP) [11, 12, 13, 14, 15] and in the susceptible-infected-susceptible(SIS) [1, 3, 4, 5, 6, 7, 10] models,mainly based on perturbative approaches around thetransition point [1, 2, 3, 4, 7, 9, 12], even though non-perturbative analyses have recentlybeenperformed[10]. Onleave at Departament de Fsica i Enginyeria Nuclear, Universitat Polit`ecnica de Catalunya,Barcelona,SpainarXiv:1402.2832v2 [cond-mat.stat-mech] 7 May 2014Heterogeneouspair-approximationforthecontactprocessoncomplexnetworks 2The heterogeneous mean-eld (HMF) approach for dynamical processes on complexnetworkshasbecomewidespreadinthelastyears. Thistheory, formerlyconceivedtoinvestigatetheSISdynamicsoncomplexnetworks[16], assumesthat thenumber ofconnections of a vertex (the vertex degree) is the quantity relevant to determine its state,neglectsall dynamical correlationsaswell astheactual structureof thenetwork. Ontheotherhand,thequenchedmean-eld(QMF)theory[3,17]stillneglectsdynamicalcorrelationsbuttheactual quenchedstructureof thenetworkisexplicitlytakenintoaccountbymeansoftheadjacencymatrixAijthatcontainsthecompleteinformationoftheconnectionamongvertices[18]. Morerecently, semi-analyticmethodsincludingdynamicaluctuations[5,6]andheterogeneouspair-approximations[8,19,20,7]haveappearedasmoreaccuratealternativestoHMFtheory.The CP[21] is the simplest reaction-diusion process exhibiting a transitionbetweenanactiveandafrozen(absorbing)phase[22]. TheCPdynamicsinvestigatedinthepresentworkisdenedasfollows[22]: Avertexi of anarbitraryunweightedgraphcanbeoccupied(i=1)orempty(i=0). Atarate, anoccupiedvertextriestocreateanospringinarandomlychosennearest-neighbor, whathappensonlyif itisempty. Anoccupiedvertexspontaneouslydisappearsatrate1(thisratexesthetimeunit). NoticethatintheSISdynamicsanoccupiedvertexcreates(infectsinthe epidemiological jargon) an ospring in each empty nearest neighbor at rate . Evenbeingequivalentforstrictlyhomogeneousgraphs(ki k i), thesemodelsareverydierentforheterogeneoussubstrates(seediscussioninRef. [23]). However, inbothmodelsthecreationof particlesisacatalyticprocessoccurringexclusivelyinpairsofempty-occupiedvertices, implyingthatthestatedevoidfromparticlesisaxedpointofthedynamicsandiscalledabsorbingstate.Afteranintensediscussion[13,12,24,25,26],theHMFtheoryshowedupasthebest available approach to describe scaling exponents associated to the phase transitionoftheCPonnetworks[27]. However,someimportantquestionsremainedunanswered.Thetransitionpoint c=1predictedbytheHMFtheory[24] does not reect thedependence on the degree distribution observed in simulations [11, 24, 27]. Mostintriguingly, it was observedagoodaccordancebetweensimulations andaheuristicmodicationof thestrictlyhomogeneouspair-approximation(HPA)(seeRef. [22] fora review) where the xed vertex degree is replaced by the average degree of thenetwork[11,27]:c=kk 1. (1)Finally, sub-leadingcorrectionstothenite-sizescaling, undetectedbytheone-vertexHMF theory, are quantitatively relevant for the analysis of highly heterogeneousnetworks, forwhichdeviationsfromthetheoretical nite-sizescalingexponentswerereported[27].Dynamicalcorrelationsrepresentanimportantfactortoascertaintheaccuracyoftheanalytical results. Thesimplestwaytoexplicitlyconsiderdynamical correlationsis by means of a pair-approximation[22]. Inthis paper, we present a pair HMFHeterogeneouspair-approximationforthecontactprocessoncomplexnetworks 3approximationfortheCPonheterogeneousnetworks. Weshowthatthistheoryyieldsgreat improvements inrelationtothe one-vertexcounterpart but, however, maybefarther fromthesimulationthresholds thanEq. (1). This apparent contradictionissolved showing that the higher-order homogeneous cluster approximations overestimatetheactualtransitionpointimplyingthattheproximityisonlyacoincidence. WealsoshowthatpairHMFtheoryyieldsthesamecriticalexponentsastheone-vertexHMFtheory,butwithdierentcorrectionstothescaling. Thesecorrectionsallowanalmostperfect match with simulations constituting a great improvement in relation to the one-vertexmean-eldtheories[15,14].The paper is organized as follows: Pair HMF theory is proposed and thetranscendental equation that gives the transition points is derived in section 2.Thenumerical analyses of thethresholds andthecomparisons withquasi-stationarysimulations are presented in section 3. The critical exponents are analyticallydeterminedandcomparedwithsimulationsinsection4. Ourconcludingremarksaredrawninsection5.2. PairHMFtheoryInthissection, wedevelopthepairHMFtheorywheretheevolutionofthesystemisgiven by the average behavior of vertices with the same degree. So, let us introduce thenotationbasedonRef. [7]: [Ak] istheprobabilitythatavertexof degreekisinthestate A; [AkBk ] is the probability that a vertex of degree kin state A is connected to avertex of degree k

in state B; [AkBk Ck ] is the generalization to three vertices such thatthepairs[AkBk ]and[Bk Ck ]areconnectedthroughanodeofdegreek

andsoforth.Anoccupiedstateisrepresentedby1andanemptyoneby0. Thepair-approximationcarried out hereafter uses the following notation: [1k] = k, [0k] = 1 k, [0k1k ] = kk ,[1k0k ] =kk , [1k1k ] =kk and[0k0k ] =kk . Obviously, wehavethatkk =k

k,kk = k

k, and kk =k

k. Independently of the dynamical rules, the following closurerelationscanbederivedfromsimpleprobabilisticreasonings:kk+ kk = k

kk+kk = kkk+ kk = 1 kkk+kk = 1 k . (2)Themasterequationfortheprobabilitythatavertexwithdegreekisoccupiedtakestheformdkdt= k + k

k

kk

k

P(k

[k), (3)wheretheconditional probabilityP(k

[k), whichgives theprobabilitythat avertexof degree k is connectedtoa vertexof degree k

, weighs the connectivitybetweencompartments of degrees kand k

. The rst term of Eq. (3) represents the spontaneousHeterogeneouspair-approximationforthecontactprocessoncomplexnetworks 4annihilation and the second term reckons the creation in a vertex of degree kdue to itsnearestneighbors. Thedynamicalequationforkk isdkk

dt= kk kk

k

+ kk+ (k

1)

k

[0k0k 1k ]P(k

[k

)k

(k 1)

k

[1k 0k1k ]P(k

[k)k

. (4)Therst termrepresents theannihilationinthevertexof degreek

, thesecondoneincludesthecreationinthevertexofdegreekduetotheconnectionwiththeneighborof degreek

andthethirdoneisduetotheannihilationof thevertexwithdegreek.Thesetermsrepresentthereactionsinsidepairswithdegreeskandk

, thatcreateordestroyaconguration[0k, 1k ]. Thefourthandfthtermsrepresentchangesduetocreationinverticeswithdegreek

andk,respectively,duetoalltheirneighborsexceptthe link between the vertices of the pair itself, which is explicitly included in the secondterm.Theone-vertexmean-eldequationproposedinRef.[24]isobtainedfactoringthejointprobabilitykk(1 k)k inEq. (3). Detailsof one-vertexsolutioncanbefound elsewhere [14]. Finally,the factor k

1 preceding the rst summation in Eq. (4)isduetothek

neighborsofmiddlevertexexceptthelinkofthepair[0k0k ](similarlyfork 1precedingthesecondsummation).WenowapproximatethetripletsinEq. (4)withastandardpair-approximation[28,29,30][Ak, Bk , Ck ] [Ak, Bk ][Bk , Ck ][Bk ], (5)tonddkk

dt= kk kk

k

+ kk+(k

1)kk

1 k

k

k

k P(k

[k

)k

(k 1)kk

1 k

k

kk P(k

[k)k

. (6)SubstitutingEqs.(2)in(6)andperformingalinearstabilityanalysisaroundthexedpointk 0andkk 0,onendsdkk

dt= _2 +k

_kk+ k+ (k

1)

k

k

k P(k

[k

)k

. (7)The next step is to perform a quasi-static approximation for t , in which dk/dt 0anddkk /dt 0,tondkk =2k

12k

+ k . (8)Finally,weplugEq.(8)inEq.(3)toproducedkdt=

k

Lkk k , (9)Heterogeneouspair-approximationforthecontactprocessoncomplexnetworks 5wheretheJacobianLkk isgivenbyLkk = kk+k(2k

1)P(k

[k)(2k

+ )k

= kk+ Ckk , (10)withkk beingtheKroneckerdeltasymbol.The absorbingstate is unstable whenthe largest eigenvalue of Lkk is positive.Therefore, thecritical point is obtainedwhenthelargest eigenvalueof theJacobianmatrix is null. Let us focus only on uncorrelated networks where P(k

[k) =k

P(k

)/k [31]. It is easy to check that uk= k is an eigenvector of Ckk with eigenvalue =k

k

(2k

1)P(k

)k

(2k

+ ). (11)Since Ckk > 0 is irreducible (all compartments have non-null chance of being connected)and uk> 0, the Perron-Frobenius theorem [18] warranties that is the largest eigenvalueof Ckk . The transitionpoint is, therefore, givenby 1+=0 that results thetranscendentequationck

k

(2k

1)k

P(k

)(2k

+ c)= 1, (12)whichcanbenumericallysolvedforanykindofnetwork(section3).Tochecktheconsistencyof thetheory, weconsidertherandomregularnetworks(RRNs) that are strictly homogeneous networks with vertex degree distribution P(k) =k,mandconnections done at randomavoiding self andmultiple edges [32]. UponsubstitutionofP(k)inEq.(12),oneeasilyshowsthatthetransitionpointisc=mm1, (13)that is thesameof thesimplehomogeneous pair-approximation. Simulations of CPonRRNs withm=6yieldacritical point c=1.2155(1) [33], slightlyabove thepair-approximationpredictionc= 1.2.3. ThresholdforarbitraryrandomnetworksInthissection, wecomparethethresholdsgivenbyEq. (12)withsimulationsof theCPdynamicsonrandomnetworksgeneratedbytheuncorrelatedcongurationmodel(UCM) [34]. Power law degree distributions P(k) k, where is the degree exponent,withminimumdegreek0andstructuralcutokc= N1/2,thelatterrenderingnetworkswithoutdegreecorrelations[31],wereused. Thischoiceissuitableforcomparisonwiththe pair HMF theory where such a simplication was adopted. We investigated networkswitheitherk0=3or6. Thelatteristocomparewiththeresultsof Ref. [27] andtoremarktheimprovementof thepresenttheory. Networksof sizesuptoN=107anddegreeexponents= 2.3, 2.5, 2.7, 3.0and3.5wereanalyzed.The thresholds for heterogeneous pair-approximations were determinedfor eachnetwork realization and averages done over 10 independent networks. Sample-to-sampleuctuations of the thresholdpositions become very small for large networks. TheHeterogeneouspair-approximationforthecontactprocessoncomplexnetworks 6thresholds against networksizefor twodegreeexponents areshowninFig. 1. TheresultsarecomparedwiththeheuristicformulainspiredintheHPAtheorygivenbyEq.(1). WeperformedsimulationsofCPdynamicsonthesamenetworksamplesused103104105106107108N1.101.151.201.251.301.35thresholdPHMFHPAHTAp103104105106107108N11.051.11.151.21.251.3thresholdPHMFHPAHTAp103104105106107108N1.201.251.301.351.401.45thresholdPHMFHPAHTAp103104105106107108N1.11.121.141.161.18thresholdPHMFHPAHTApFigure1. ThresholdsagainstnetworksizefortheCPonUCMnetworkswithdegreeexponents=2.50(top)and=3.50(bottom), k0=3(left)andk0=6(right),obtained in mean-eld theories and QS simulations (position of the susceptibility peakp). Dashedlinesarenon-linearregressions,Eq.(17),usedtoextrapolatetheinnite-sizelimitofthethresholds. Acronyms: PHMF(pairheterogeneousmean-eld, HPA(homogeneouspairapproximation),HTP(homogeneoustripletapproximation).toevaluatethemean-eldtheories. Thestandardsimulationschemewasused[22]: Anoccupiedvertexj is randomlychosen. Withprobabilityp=1/(1 + ) theselectedvertexbecomesvacant. Withcomplementaryprobability1 poneof thekjnearest-neighborsofjisrandomlychosenand,ifempty,isoccupied. Thetimeisincrementedbyt=1/[(1 + )n(t)], wheren(t)isthenumberofparticlesattimet. Toovercomethe diculties intrinsic to the simulations of systems with absorbing states [22], we usedthequasi-stationary(QS)simulationmethod[35],inwhicheverytimethesystemtriestovisitanabsorbingstateit jumpstoanactivecongurationpreviously visitedduringthesimulation(anewinitial condition). Details of themethodwithapplications todynamicalprocessesonnetworkscanbefoundelsewhere[15,33].The QS probabilityP(n), dened as the probability that the system has n occupiedvertices in the QS regime, was computed after a relaxation tr= 106during an averagingHeterogeneouspair-approximationforthecontactprocessoncomplexnetworks 7time ta=107. The transitionpoint for nite networks was determinedusing themodiedsusceptibility[32] n2 n2n=N(2 2), (14)whichisexpectedtohaveadivergingpeakthatconvergestothetransitionpointwhenthenetworksizeincreases.The choice of the alternative denition, Eq. (14), instead of the standardsusceptibility = N(2 2)[30]isduetothepeculiaritiesofdynamicalprocessesoncomplexnetworks. Forexample, theCPonannealednetworks, forwhichtheQSprobabilitydistributionatthetransitionpointhastheanalyticallyknownform[15]P(n) =1f_N_, (15)where =N/g, g =k2/k2andf(x) is ascalingfunctionindependent of thedegreedistribution. Itiseasytoshow[23] that nl l, leadingto and /N g1. Usingthescalingpropertiesofg[31],g _k3c= N(3)/2 < < 3const. > 3, (16)forcutoscalingaskc N1/, oneconcludesthat, at=c, Nand N

where=min[( 3 + )/2, 1/2] >0and

=min[( 3)/, 0] 0. So, thesusceptibilityalwaysdivergesatthetransitionpointwhile doesnot.Typical susceptibility versus curves are shown in Fig. 2. The peak positions shiftleftwards convergingtoanitethresholdas networksizeincreases. Noticethat thelarger thedegreeexponent thenarrower thesusceptibilitycurves andthefaster theconvergencetotheasymptoticthreshold. Theinnite-sizethresholdcisestimatedinQSsimulationsaswellasinthemean-eldtheoriesusinganextrapolationc(N) = c + a1Nb1(1 + a2Nb2). (17)AsonecanseeinFig.1,thecurvescvs. Nfordierentmean-eldtheoriesareonlyshiftedindicatingthattheexponentsbiarethesame. Theycanbeobtainedusingacontinuousapproximationk =_kck0kP(k)dk 1 2k0_1 (kc/k0)2(18)inEq. (1)toobtainb1=b2=( 2)/forkc N1/, where=max(2, 1)forUCM networks [34]. These biexponents can also be derived directly from equation (12)inamorecomplexcalculationthatisomittedforsakeofbrevity. Inannealednetworks, thevertexdegreesarexedwhiletheedgesarecompletelyrewiredbetweensuccessivedynamicsstepsimplyingthatdynamical correlationsareabsentandHMFtheorybecomesanexactprescription[14].Heterogeneouspair-approximationforthecontactprocessoncomplexnetworks 8 k0= 3 k0= 6PHMF cPHMF c2.30 1.098(1) 1.1009(5) 1.043(1) 1.044(1)2.50 1.1415(4) 1.1473(6) 1.0628(8) 1.0641(5)2.70 1.1817(3) 1.1906(4) 1.0788(4) 1.0810(7)3.00 1.2320(3) 1.2479(3) 1.0977(2) 1.1011(4)3.50 1.2938(1) 1.3224(2) 1.1200(1) 1.1248(4)Table 1. Transitionpoints of the CPonUCMnetworks withdierent degreeexponents,minimumvertexdegreek0= 3ork0= 6,andstructuralcutokc= N1/2forpairheterogeneousmean-eld(PHMF)theoryandQSsimulations(c). Numberinparenthesisareuncertaintiesinthelastdigit.TheexponentsbiinQSsimulationsdierfromthoseof themean-eldtheories.TheycanbeanalyticallyestimatedusingthescalingtheorypresentedinRefs.[14,15].TheQSdensityatthetransitionpointscalesas (c) (gN)1/2(19)whileaboveit ( c), (20)where=max[1, 1/( 2)] [14]. ThesescalinglawsareconrmedinthepairHMFtheorydevelopedinsection4. Assumingthatbothscalinglawsholdatponeobtainspc (gN)1/2. (21)Usingagainthecontinuousapproximationtocomputegandneglectinghigherordertermsonendsg= C

_3[1 + 2+] < 31 3+ > 3(22)where, kc/k0, C= [( 2)2/(3 )( 1)[ and a logarithmic dependence is foundfor = 3. Upon substitution of gin Eq. (21), the exponents b1= ( 2)(3 + )/2andb2=( 2)/for 3andkc N1/arefound.Weperformednon-linear regressionsusingEq. (17) withc, a1anda2freeandxingbiaccordingtothetheoretical corrections. Excellenttswereobtained, ascanbeseeninFig.1andthenumericalestimatesofcareshownintable1. Asexpected,pair HMFtheoryis averygoodimprovement whencomparedwiththe one-vertexapproximation c= 1. However, for some values of the heuristic HPA theory is closerto simulations than the pair HMF theory,as canseen inFig. 1. It is a surprising resultsince heterogeneity is expected to play an important role in dynamical correlations evenfordegreedistributionswithoutaheavytailasinthecase> 3.The puzzle behind this apparent paradox is that cluster approximationsunderestimate the real thresholdandthe convergence is expectedonlyinthe limitHeterogeneouspair-approximationforthecontactprocessoncomplexnetworks 91 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45101102103c[=2.30]c[=3.50]1 1.05 1.1 1.15 1.2 1.25101102103c[=2.30]c[=3.50]Figure 2. Susceptibilityagainst creationrate for =2.30(leftmost curves) and=3.50(rightmostcurves), k0=3(left)andk0=6(right). ThenetworksizesareN= 104, 3 104, 105, 3 105, 106, 3 107, 107,increasing from the right. Dashed linesaretheextrapolationsofthepeakpositionsforN .of largeclusterapproximations. Ahomogeneoustripletapproximation(HTA)fortheCPonunclusterednetworksyieldsthethreshold[33]:c= k + 2_k2k3k 4. (23)Comparing this approximation with simulations, gure 1, one sees that HTA thresholdsare,asexpected,higherthantheHPAonesbutoverestimatethesimulationthresholdsfor all investigatednetworks, moreevidentlyfor k0=3. Thisresult showsthat thehomogeneous cluster approximations will converge to a threshold above the correct oneand they are, in principle, not applicable to the CP dynamics on heterogeneous networksas previouslydone[11, 27]. TheproximitybetweenHPAtheoryandsimulations isthereforeacoincidence.4. CriticalexponentsIn this section, the critical exponents of the CP in the pair HMF theory are derived andcomparedwithresultsofQSsimulations.4.1. Critical exponentsinthepairHMFtheoryforinnitenetworksItiswellknownthatclusterapproximationsofhigherordersimprovethecriticalpointestimates but dochangethecritical exponents inlatticesystems [30]. As expected,thepairHMFtheoryfortheCPyieldsthesamescalingexponentsastheone-vertexapproximation[12,15,14],changingonlytheamplitudesandthenite-sizecorrectionstothescalingaswewillshowinthissection.Ina pair level, the scaling exponents associatedto the absorbing state phasetransitioncanbe derivedfromEqs. (3) and(6) keepingterms uptosecondorder.Heterogeneouspair-approximationforthecontactprocessoncomplexnetworks 10Assumingagainuncorrelatednetworks,thedynamicalequationsbecomedkdt= k +kk

k

kk P(k

) (24)anddkk

dt= kk kk

k

+ kk+(k

1)k(1 + k kkk )

k

k

k P(k

)(k 1)kkk

k

kk P(k

) +O(3). (25)Thequasi-staticapproximationwithdk/dt 0anddkk /dt 0leadstokk =2k

12k

+ k

_1 +( + 1)(k

1)(2k

1)(2k

+ )

k_k

12k

1+k

(k 1)k(2k

+ )_k_+O(3),(26)whichisinsertedinEq.(24)toresultdkdt= k +kk_1k_23k__(27)and,consequently,thestationarydensityk=k1/k1 + k2/k 3/k, (28)whereiaregivenby1= (+1)

k_P(k)k(2k + ) P(k)(k 1)2k(2k + )2_=/1() +a1()2, (29)2= ( + 1)

kP(k)(3k + )k(2k + )2= /2()/1() + a2()2+ (30)and3=

kP(k)(2k 1)kk(2k + )2= /3()/1() + a3()2, (31)where =

k P(k)kwhileAiareconstantsoforder1givenby/1() = 1 +( + 1)k

kkP(k)2k + , (32)/2() = 1 ( + 1)k

kk2P(k)(2k + )2(33)and/3() =k

kk2(2k 1)P(k)(2k + )2. (34)TherightmostsidesofEqs. (29)-(31)wereobtainedusingi