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Resilience to damage of graphs with degree correlations 1 2 Alexei Va´zquez and Yamir Moreno 1INFN and International School for Advanced Studies, Via Beirut 4, 34014 Trieste, Italy and 2The Abdus Salam International Centre for Theoretical Physics, Condensed Matter Group, P.O.Box 586, Trieste I-34014, Italy. (Dated: February 1, 2008) 3 The existence or not of a percolation threshold on power law correlated graphs is a fundamental 0 questionforwhichageneralcriterionislacking. Inthisworkweinvestigatetheproblemsofsiteand 0 bondpercolationongraphswithdegreecorrelationsandtheirconnectionwithspreadingphenomena. 2 Weobtainsomegeneralexpressionsthatallowthecomputationofthetransitionthresholdsortheir n bounds. Using these results we study the effects of assortative and disassortative correlations on a theresilience to damage of networks. J 7 PACSnumbers: 89.75.Hc,05.20.-y,89.75.-k,02.50.-r ] n The graphs representing many real networks are char- oredgeremoval)ofrandomgraphswitharbitrarydegree n acterized by power law degree distributions [1]. The ori- distributions andcorrelationsby addressingthe problem - s ginofthesepowerlawscanbetracedbacktothegrowing of dilute (site or bond) percolation on these graphs. We i d natureofrealnetworksandtosomeeffectivepreferential reportageneralequationforthethresholdandboundit. . attachment mechanism. This later mechanism implies Besides, we analyze the effect of correlations considering t a that when new vertices are added to the graph they are some examples of uncorrelated,assortativeanddisassor- m more likely linked to existing vertices with large degrees tative correlated graphs or their mixture. We conclude - [1, 2]. Recently, there has been a great interest in the that assortative correlations can make graphs quite ro- d 2 n study of processesrunning on top ofthese graphs due to bust, even with a finite d . On the contrary, disassor- o their social, technological and scientific relevance. Per- tative correlations can m(cid:10)ak(cid:11)e graphs fragile, even with a c colation processes [3, 4], spreading phenomena [5, 6, 7], divergent d2 . [ the Isingmodel[8]andsearchingtechniques[9]aresome Letusst(cid:10)art(cid:11)byconsideringthesetofundirectedgraphs 3 examples for which analytical solutions have been found withN verticesandarbitrarydegreedistributionp . Fol- d v in random graphs with the only constraint given by the lowingone endofa randomlychosenedge,we willfind a 2 degree distribution. One of the fundamental results is vertexofdegreedwithprobabilityq =dp /hdi. Wefur- 8 d d that the threshold characterizing the percolation tran- ther assume correlations between adjacent vertices: The 1 9 sition or an epidemic outbreak, depends on the ratio conditional probability p(d′|d) that a vertex of degree d′ 2 0 d /hdiofthefirsttwomomentsofthedegreedistribu- isreachedfollowinganyedgecomingfromavertexofde- 2 t(cid:10)ion(cid:11)[3, 4, 5, 8]. Hence, if d2 diverges when increasing gree d explicitly depends on both d and d′. Consistency 0 ′ at/ tnhaemgicralipmhits.ize, there is no(cid:10) tr(cid:11)ansition in the thermody- wBeitshideths,ethdeegjroeinetdpisrtorbibabuitliiotny pr(edq′u|dir)eqsd Pthadt′pt(hde|tdw)o=ve1r-. m The topology of real networks is also characterized by ticesateitherendofarandomlychosenedgehavedegrees ′ d and d must be symmetric. For uncorrelated networks - degree correlations [10, 11] and, therefore, the extension ′ d of previous results for uncorrelated graphs is of utmost p(d|d)=qd′ that is independent of d. n Theproblemofpercolationongraphswithdegreecor- importance. Moreover, it has been shown that growing o relations has been recently studied [11] using the gen- c network models with [12] and without [13] preferential erating function formalism. Alternatively, one can use : attachment lead to non-trivial degree correlations. The v a more general statistical mechanics approach [16]. In study of models on graphs with degree correlations is i X this case the size of the giant component, the fraction of quite recent [11, 14, 15, 16]. Some expressions for the r size of the giant component and related quantities have nodes in the largest cluster, is given by a been obtained in Ref. [11] whereas an equation for the S =1− p (u )d, (1) epidemic threshold has been provided in [14]. General d d statisticalmechanicsapproachesformodelsoncorrelated Xd graphs has also been developed in Refs. [15, 16]. How- egvenere,rainl ccroitnetrriaosntftoor tthhee ecxaisseteonfceunocronrroetlaotfeda tgrraanpshitsi,ona ud = p(d′|d)(ud′)d′−1, (2) threshold has not been proposed yet. A first step in this Xd′ directionhasbeentakeninRef. [17]foradiseasespread- where u is the average probability that an edge con- d ing model. nected to a vertex of degree d leads to another vertex Inthispaperwestudytheresiliencetodamage(vertex that does not belong to the giant component [11]. 2 Let us generalize this result to the site percolation on a simple topological measure, whether or not a given problem. In this case a fraction f of the nodes is re- graph is robust under vertex removal. moved from the graph and the new giant component is In the bond percolation problem a fraction f of the computed. Since the node removalis independent of the edgesis removedfromthe graphandthe new giantcom- node degree this is equivalent to replace the original de- ponent is computed. Since the edge removal is made gree distribution and correlations by: (i) the probability at random this is equivalent to keep the original degree that a node selected at random has degree d and it has distribution and replace the degree correlations by the not been removed,and (ii) the probability that if we se- probabilitythatifweselectanodeatrandomandfollow ′ lect anode atrandomandfollow one ofits edgeswe end one ofits edges we endin a node with degreed that has in a node with degree d′ that has not been removed, i.e. not been removed, i.e. pd →(1−f)pd, p(d′|d)→(1−f)p(d′|d), (3) pd →pd, p(d′|d)→(1−f)p(d′|d). (11) Substituting Eqs. (3) in Eqs. (1) and (2) we get Substitution of Eq. (11) in Eqs. (1) and (2) yields S =1−f −(1−f) pd(ud)d, (4) S =1− p (u )d, (12) d d Xd Xd ud =f +(1−f)Xd′ p(d′|d)(ud′)d′−1, (5) ud =f +(1−f)Xd′ p(d′|d)(ud′)d′−1. (13) where the term −f (f) in Eq. (4) ( Eq. (5)) gives the Note that the only difference between the site and bond probability of hitting a removed node. One solution to percolation problems (see Eqs. 4,5) is the equation for these equations is u = 1 yielding S = 0. This solution d thegiantcomponentwhilethatforu isidentical. Hence, is valid whenever the equation for the u is stable un- d d Eqs. (8) and (9) are also valid for the bond percolation der successive approximations. That is, if we start with problem. u (n) = 1−ρ (n) and compute the successive approxi- d d In what follows we consider some particular graphs in mation ρ (n+1) then we should obtain that ρ (n) →0 d d order to analyze the effects of correlations. Depending in the limit t → ∞. For ρ (n) ≪ 1 the last equation is d onthe monotony ofhdi (d) the degreecorrelationscan approximated by the linear map nn be classified in: uncorrelated if it is independent of d, assortativeorpositiveifitincreaseswithincreasingdand ρd(n+1)= Ldd′ρd′(n), (6) disassortative or negative if it decreases with decreasing Xd′ d. A similar definition has been introduced in Ref. [11] with using a correlation coefficient. ′ ′ For random graphs with no constraint other than the Ldd′ =(1−f)Cdd′, Cdd′ =(d −1)p(d|d). (7) oneimposedbythedegreedistributionwehavep(d′,d)= The stability of the solution u = 1 is then related to qd′. In this case the lower and upper bounds in Eq. (9) d are equal giving for the largest eigenvalue thelargesteigenvalueofLdd′. Ifitissmaller(larger)than 1 the solution is stable (unstable). SinceLdd′ is linear in d2 f the stability condition can be written as Λunco = −2. (14) max (cid:10)hdi(cid:11) f >f , (1−f )Λ =1, (8) c c max AlternativelyonecancomputeΛdirectlyfromtheeigen- where Λmax is the largest eigenvalue of Cdd′ provided value problem of Cdd′. Then from Eq. (8) we obtain 2 that Λmax > 1. If Λmax < 1 the graph does not have a 1 − fc = 1/( d /hdi − 2) [4]. Hence, if the second giant component even for f = 0. Moreover, since Cdd′ moment d2 (cid:10)div(cid:11)erges the threshold equals 1, i.e. the is a positive matrix then Λmax has the lower and upper network i(cid:10)s ro(cid:11)bust under random vertex or edge removal. bounds mind d′Cdd′ and maxd d′Cdd′, yielding Furthermore, consider the case in which the degree cor- P P relations can be decomposed into two components min hdi (d)≤1+Λ ≤max hdi (d). (9) d nn max d nn ′ ′ p(d|d)=αqd′ +(1−α)δp(d|d) (15) where ′ ′ ′ ′ with 0 < α < 1 and δp(d|d) > 0 for all (d,d). Vary- hdi (d)= p(d|d)d, (10) nn ing the parameter α one interpolates between the un- Xd′ correlated graphs (α = 1) and a graph with arbitrary ′ istheaveragedegreeamongtheneighborsofanodewith degree correlations given by δp(d|d). In this case from degree d [10]. Eq. (9) can be used to determine, based Eq. (9) we obtain Λ ≥ αΛunco and, therefore, if the max max 3 100 and d diverges f = 1. Therefore, we can conclude max c that the divergence of the second moment is not a nec- 10−2 essary condition. Letusnowanalyzeifthedivergenceofthesecondmo- 1 0) 10−4 mentisasufficientconditionforfc =1,usinganexample ()/(SfS 10−6 Λ/dmaxmax0.5 odgf>aa1dvieasrantsdesxoaranttaettdihvgeeeogitnrhacepidrhee.nnCtdotionssicitdh.eoTrsehanevnaetrwtreiatxhndwpoirtmohbadambegiolrinetyge d 10−8 0 all vertices with degree d′ > 1, otherwise it is connected 0 0.5 1 α to a vertex with d′ =1 chosen at random, i.e. 10−10 0 0.2 0.4 0.6 0.8 1 ′ f ′ (1−gd′)dpd′ ′ p(d|d) = Θ(d −1)δd,1 (1−g )sp s s s + (P1−gd)δd′,1Θ(d−1) cFdI−G3..51:(2S≤izedo≤f tdhmeagxi,andtmcaoxm=po1n0e0n)tafnodr adeggrraepehcowrirtehlatpidon=s + gd gd′d′pd′ Θ(d′−1)Θ(d−1). (19) p(d′|d) = αδdd′ +(1−α)qd′ with α, as computed from Eq. sgssps (4). The dashed line marks the percolation threshold ob- P tainedusingperturbationtheory(Eq. (18)). Theinsetshows where Θ(x) is the unitary step function (Θ(x) = 0 for thelargesteigenvaluerelativetodmax asafunctionofα. The x ≤ 0 and Θ(x) = 1 for x > 0). Moreover, the fraction pointswerecomputed numerically and thelineis thepertur- of nodes with degree 1 is obtained self-consistently from bation theory dependencyΛmax/dmax =α. theconditionp1 = d>1(1−gd)dpd. Theaveragedegree of the neighbors ofPa node with d>1 is given by nbeetwroobrukstisforrobaunsytαfo>r t0h.eTuhnicsoirmremlaetdeidatcealyseimitpwlieilsltahlasot hdinn =1+gd(cid:18)Pd′>1ggd′sdp′2pd′ −1(cid:19), (20) any graph with a divergent second moment and a finite s s s P amount of random mixing of edges does not have a per- and, therefore, these graphs are disassortative for any colation threshold. monotonicdecreasingfunctiong . To analyzethe perco- d Assortative correlations allow us to show that the di- lation properties of this graph we computed exactly the vergence of the second moment is not a necessary condi- largest eigenvalue of Cdd′ =(d′−1)p(d′|d), resulting tion for the absence of the threshold. Let us consider a network with degree correlations g2(d−1)dp Λ = d d d. (21) max P g sp p(d′|d)=αδdd′ +(1−α)δp(d′|d), (16) Ps s s Hence,theconditionsforthe existenceofagiantcompo- ′ ′ with 0 < α < 1 and δp(d|d) > 0 for all (d,d). α = 1 nent(Λ >1)orresiliencetodamage(Λ =∞)are max max correspondsto a fully assortativegraphmade up of sub- modulatedby g and,therefore,the disassortativecorre- ′ d graphs with fixed degree. In this case Cdd′ = dδdd′ (see lations given by g have a great impact on the percola- d Eq. 7) is already diagonal. The largest eigenvalue is tion properties. For instance, let us consider g = d−α d Λmax =dmax, where dmax is the largest degree. If dmax and a power law degree distribution p = cd−γ with d diverges for N → ∞ then fc = 1. For the more general γ < 3 ( d2 = ∞). From Eq. (20) it follows that case 0 < α < 1 we compute the largest eigenvalue using hdi −1(cid:10)∼(cid:11)d−α so that when increasing α the graph perturbation theory [18] around α=1, obtaining nn gets more and more disassortative. Moreover, Λ di- max verges for α < α = (3−γ)/2 and it is finite otherwise. Λ (α)=αd +(1−α)C . (17) c max max dmaxdmax Thus, for small values of α the graph is robust but for α > α it becomes fragile. It is worth noticing that the Thisresultisvalidwhenever(1−α)C ≪αd . c dmaxdmax max value of α above which the giant component disappears In generalC decreaseswith increasingd , re- dmaxdmax max (Λ <1) is larger than α . Besides, for large degrees, sulting max c thedegreedistributionoftheverticesinthegiantcompo- Λ (α)≈αd , (18) nentisstillapowerlaw,butitdecaysslowerthanthatof max max the whole graph. Thus, disassortative correlations com- for d ≫ 1/α. Hence, for any α > 0 and any un- petes againstthe formationof the giantcomponent and, max boundeddegreedistributionwe havef =1,i.e. there is the divergence of d2 is not a sufficient condition to get c no percolation threshold. In Fig. 1 we show the validity a robust graph wi(cid:10)th f(cid:11)c =1. of the perturbation theory for a particular perturbation The connection between percolation theory and mod- ′ δp(d|d). Thus, as in the fully assortative case, if α > 0 elsofepidemicspreadingiswellknown[19]. Twogeneral 4 classesofepidemiologicalmodelscanberelatedtoperco- expressions to obtain or bound the transition threshold. lationproblems,theSusceptible-Infected-Removed(SIR) Using these results we have shown that the existence of and the Susceptible-Infected-Susceptible (SIS) classes. a finite amount of randommixing of the connections be- The SIR model assumes that individuals can exist in tween vertices is sufficient to make the graph robust un- 2 three classes and that once they get infected they can der vertex or edge removal provided d → ∞. Assor- notcatchtheinfectionagain. Thismodelcanbemapped tative correlations makes the situation(cid:10) ev(cid:11)en better, they intoabondpercolationproblemtakingf astheprobabil- can lead to a graph robust to randomdamage even with ity thatthe diseasewillbe transmittedfromone node to a finite second moment of the degree distribution. On anotherandthesizeofthegiantcomponentasthesizeof the contrary, disassortative correlations compete again theoutbreak. Hence,alltheconclusionsdrawnabovefor the formation of the giant component and can make a the bond percolation problem can be translated to the graph fragile even with a divergent second moment. languageofepidemicspreadingfortheSIRmodelontop We thank A. Vespignani, R. Pastor-Satorras and M. ofgraphswithdegreecorrelations,extending inthis way Weigt for helpful comments. This work has been par- previous studies in Refs. [6, 7] for uncorrelated graphs. tiallysupportedbytheEuropeancommissionFETOpen On the other hand, the SIS model allows individuals project COSIN IST-2001-33555. to movethroughthe cycle of infection so that the prevalence (number of infected individuals) attains a station- ary value. The SIS model on top of graphs with degree correlationshas been recently analyzedin Refs. [14, 17]. They obtained the epidemic threshold (the value of λ [1] R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74, 47 above which the solution with zero prevalence is unsta- (2001). ble)λ=1/Λ′ ,whereΛ′ isthelargesteigenvalueof [2] S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. 51, max max the matrix C′ = dp(d′|d). This approach is quite sim- 1079 (2002). dd′ ilar to the one presented here for site percolation with [3] R. Albert, H. Jeong, and A.-L. Barabási”, Nature 406, ′ 378 (2000); M.E.J. Newman, S.H. Strogatz, and D.J. the remark that C is different (see Eq. (7)). In fact, dd′ ′ ′ Watts, Phys. Rev. E 64, 026118 (2001); D. S. Callaway, if y is an eigenvector of C = dp(d|d) correspond- d dd′ M. E. J. Newman, S. H. Strogatz, and D. J. Watts, ′ ing to the eigenvalue Λ then yd/d is an eigenvector of Phys.Rev. Lett. 85, 5468 (2000); R. Cohen, K. Erez, ′′ ′ ′ Cdd′ = dp(d|d) corresponding to the same eigenvalue. D. ben-Avraham, and S. Havlin, Phys. Rev. Lett. 86, ′ This last matrix is that of Eq. (7), but replacing d 3682 (2001). by d′ − 1. However, this subtle difference makes the [4] R. Cohen, K. Erez, D. ben-Avraham, and S. Havlin, SIS and dilute percolation different. We have computed Phys. Rev.Lett. 85, 4626 (2000). ′′ [5] R. Pastor-Satorras, and A. Vespignani, Phys. Rev. Lett. thelargesteigenvalueofC forthedisassortativegraph dd′ 2 86, 3200 (2001); R. Pastor-Satorras, and A. Vespignani, considered above(Eq. (19)). Taking the limit d ≫ 1 Phys.Rev.E63,066117(2001);A.L.Lloyd,R.M.May, one gets (cid:10) (cid:11) Science 292, 1316 (2001). 2 [6] Y. Moreno, R. Pastor-Satorras, and A. 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