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2 Core percolation in random graphs: 0 0 a critical phenomena analysis 2 n a M. Bauer and O. Golinelli, J 7 email: bauer, golinelli @spht.saclay.cea.fr { } ] Service de Physique Théorique, Cea Saclay, 91191 Gif-sur-Yvette, France h c e January 31, 2001; revised June 22, 2001. m Preprint T01/017;arXiv:cond-mat/0102011. - t a Eur. Phys. J. B 24, 339–352(2001). t s . t a Abstract m We study both numerically and analytically what happens to a random graph of average connectivity - d α when its leaves and their neighbors are removed iteratively up to the point when no leaf remains. The n remnant is made of isolated vertices plus an induced subgraph we call the core. In the thermodynamic limit o of an infinite random graph, we compute analytically the dynamics of leaf removal, the number of isolated c vertices and the number of vertices and edges in the core. We show that a second order phase transition [ occurs at α = e = 2.718...: below the transition, the core is small but above the transition, it occupies a 2 finite fraction of the initial graph. The finite size scaling properties are then studied numerically in detail in v the criticalregion,andweproposeaconsistentsetofcriticalexponents,whichdoesnotcoincidewiththe set 1 of standard percolation exponents for this model. We clarify several aspects in combinatorial optimization 1 0 and spectral properties of the adjacency matrix of random graphs. 2 Keywords: randomgraphs,leafremoval,corepercolation,criticalexponents,combinatorialoptimization, 0 finite size scaling, Monte-Carlo. 1 0 / t a 1 Introduction of finite connected components with one closed loop. Ob- m viously, each tree shrinks to a single isolated point after - What remains of a graph when leaves are iteratively re- leaf removal. However, at α = 1 the percolation transi- d n moveduntil none remains ? The answerdepends onwhat tionoccursandwhenα>1,arandomgraphconsistsofa o is meant by leaves. forest plus a finite number of components with one closed c In the most standard definition, a leaf is a vertex with loop, plus a “giant”connectedcomponent containinga fi- : v exactly one neighbor, and leaf removal deletes this vertex nite fraction of the vertices and an extensive number of i and the adjacent edge. In the context of large random loops. Noloopis destroyedbyleafremovalsothatthe gi- X graphs where the connectivity α (the average number of antcomponentleadsto amacroscopicconnectedremnant r a neighborsofavertex)iskeptfixedandthenumberofver- after leaf removal. The percolation transition was dis- tices N , the answer is well known and interesting. covered and studied by Erdo¨s and Rényi in their seminal → ∞ When α < 1, the remnant after leaf removal is made of paper [1]. This has initiated a lot of work on the random O(N)isolatedvertices,plusasubgraphofsizeo(N)with- graph model, and many fine details concerning the struc- out leaves. When α>1, the remnant still contains O(N) tureofthepercolationtransitionhavebeencomputed(see isolated points, but the rest is a subgraph of size O(N), e.g Ref. [2]). whichisdominatedbyasingleconnectedcomponentusu- The random graph model is believed to be essentially ally called the backbone. The a priori surprising, but equivalent to a mean field approximation for percolation rather general, fact that backbone percolation and stan- on (finite dimensional) lattices, leading to critical expo- dardpercolationoccuratthesamepoint,namelyatα=1, nents which are valid above the upper critical dimension, hasaverysimple explanationforrandomgraphs. Indeed, which is d =6 for percolation. a large random graph of average connectivity α < 1 con- c sists of a forest(union offinite trees) plus a finite number Inthispaper,weconsidertheremovalofaslightlymore 1 M. Bauer and O. Golinelli — Core percolation in random graphs: a critical phenomena analysis 2 complicatedpattern: we removeateachstep notonly the Carlosimulations ofleafremovalprocesses,whichwe also leaf but also its neighbor (and consequently all adjacent useforthefinitesizescalinganalysisinthecriticalregion. edges). To avoidcumbersomecircumlocutions, inthe rest Thisleadsustothedefinitionandnumericalevaluationof of this paper, we call leaf the pair “standard leaf + its many new critical exponents. In particular, we give good neighbor”. Now leaf removal deletes two vertices (a ver- evidence that the core percolation exponents (at α = e) tex with a single neighbor and this neighbor) and all the are not the same as the critical exponents of standard edges adjacent to one or both vertices. It is quite natural percolation (at α = 1). Even if core percolation on a to study the removal of these patterns because it has a random graph can presumably be seen as a mean field numberofapplicationstographtheory: severalnumerical approximationforcorepercolationon(finitedimensional) characteristicsofagraphbehavenicelyunderleafremoval. lattices with impurities, the corresponding effective field Onesuchcharacteristic,whichwasouroriginalmotivation theory and its upper critical dimension are not known to from physics, is the multiplicity of the eigenvalue 0 in the us. adjacency matrix of the graph. Others are the minimal Insection6,wegivetwoapplicationsofourresults. We sizeofa vertexcoverandthe maximalsize ofanedge dis- showinparticularinsection6.1thatforanyαthecoreofa joint subset (the matching problem), questions which are randomgraphonlycarriesasmallnumberof0eigenvalues related to various combinatorialoptimization problems. of the adjacency matrix and that the emergence of the The matching problemhadalreadyledmathematicians core has a direct impact on the localized and delocalized to a thorough study of leaf removal (see Refs. [3, 4] and eigenvectors with eigenvalue 0. In section 6.2, we show references therein). In fact, parts of our analytical results that for α < e, the problem of finding minimal vertex have already been obtained in this context. However, we coversormaximaledgedisjointsubsets(matchings)canbe have obtained them independently by a direct enumera- handled very simply in polynomial time (in fact, in linear tion technique which turned out to be quite similar to a time once the graphis encodedinasuitableform). While counting lemma for bicolored trees that appeared in [5]. the matching problemcanalwaysbe solvedinpolynomial The main result on the structure of the remnant af- time, the minimal vertex cover problem is believed to be ter iterated leaf removal when the graph is a large ran- NP-hard for general graphs, and the same should be true dom graph of finite connectivity α is the following. The on the core of a random graph for α>e. residue consists of i(α)N +O(1) isolated points and an The formal proof that the core is a well-defined object induced subgraph without leaves or isolated points which is given in appendix. we call the core. It contains c(α)N +O(1) vertices and l(α)N+O(1)edges. Forα<e=2.718...,c(α)=l(α)=0 2 General definitions so the core is small. A second order phase transition occurs at α = e and for α > e, c(α) and l(α) are > 0. We shall argue that the core is made of a giantconnected We start with a few standard definitions. This section “core”componentplusafinitenumberofsmallconnected should be used only for reference. components involving a total of o(N) vertices. The func- Graph: A graph (also called a simple undirected graph tioni(α)isalwaysnon-vanishing,butitisnon-analyticat in the mathematical literature) G is a pair consisting of α=e. a set V called the set of vertices of G and a set E called Thephasetransitionatα=ewasfoundinitiallyforthe the set of edges of G, whose elements are pairs of distinct matchingproblem[3,4]. Physicistshoweverhaveobserved elements of V. If v,w is an edge, the vertices v and w { } independently that some properties of random graphsare arecalledadjacent orneighbors. They are the extremities singularatα=e(see Ref.[6]forreplicasymmetrybreak- of the edge v,w . Note that there is at most one edge { } ing in minimal vertex covers, Ref. [7] for a localization betweentwovertices,andthatthereisnoedgeconnecting problem, and, in a numerical context, Ref. [8] where an a vertex with itself: the wordsimple aboverefers to these anomaly close to the eigenvalue 0 in the spectrum of ran- two restrictions. dom adjacency matrices was observed). Adjacency matrix: The adjacency matrix of a graph G is a square matrix M indexed by vertices of G and v,w The paper is organized as follows. The general defini- such that M = 1 if v,w is an edge of G and 0 oth- v,w { } tionshavebeenregroupedinsection2. Theyarestandard erwise. Note that M is a symmetric 0 1 matrix with − and should be used only for reference. zeroesonthediagonal. Conversely,anysuchmatrixisthe In section 3 we define leaf removal, leaf removal pro- adjacency matrix of a graph. We denote by Z(G) the di- cessesobtainedbyiterationofleafremovalsandthe“core” mensionofthekernel(thatis,thesubspaceofeigenvectors for an arbitrary graph. with eigenvalue 0) of the adjacency matrix of G. Section4 presentsourderivationofthe mainresultsfor Induced subgraph: If V′ V, the graph with ver- ⊂ large random graphs. The analytical formulæ for i(α), tex set V′ and edge set E′ those edges in E with both c(α) and l(α) are given. extremities in V′ is called the subgraph of G induced by In section 5, these formulæ are checked against Monte- V′. M. Bauer and O. Golinelli — Core percolation in random graphs: a critical phenomena analysis 3 t u x t u x Random graph in the microcanonical ensemble: If V = 1, ,N , there are N(N−1)/2 graphs with v { ··· } L vertex set V and L edges (making a total of 2N(N−1)/2 (cid:0) (cid:1) graphs with vertex set V). Saying that all N(N−1)/2 are L equiprobable turns the set of graphs on N vertices and L w y z y z (cid:0) (cid:1) edgesintoaprobabilityspacewhoseelementswecallran- dom graphs in the microcanonical ensemble. This is the Figure 1: In this example, the leaf (v,w) is removed, as ensemble we use below for numerical simulations. well as the four edges touching v: the new graph G′ is the Random graph in the canonical ensemble: Given subgraph of Ginducedby thefiveremainingvertices. Note a number p [0,1], we introduce N(N−1) independent ∈ 2 that the vertex t is now isolated, and that a new leaf (z,y) random variables ε , 1 i < j N, each taking value i,j ≤ ≤ has been created. 1 with probability p and 0 with probability 1 p. Saying − that i,j is an edge of G if and only if ε = 1 turns i,j { } the set of all 2N(N−1)/2 graphs with vertex set V into a leaves, stop. Else, choose a leaf (v′,w′) and remove it. probability space whose elements we call random graphs This operation is iterated until no leaf remains. in the canonical ensemble. This is the ensemble we use below for analytical computations. History: A sequence G,(v,w),G′,(v′,w′), associ- Connectivity,α: Inthesequel,weareinterestedinthe ated to a step by step leaf removal process is··c·alled an large N limit with a finite limit α for 2L (microcanonical history. N ensemble) or for p(N 1) (canonical ensemble). The pa- − Isolated points, I; Core of a graph, C: The last rameterαistheaverageconnectivity,theaveragenumber term in anhistory starting fromG is a graphwhich splits of neighbors of a given vertex in the random graph. into a collection of isolated points I, and an induced sub- If N(p(1 p))1/2 and L pN(N−1), the ther- − → ∞ ∼ 2 graph C of G without leaves or isolated points which we modynamicpropertiesofaG(N,L)inthe microcanonical call the core of G. We denote the number of points in the ensemble and of a G(N,p) in the canonical ensemble are core by N and the number of edges in the core by L . c c thesame. ThisisinparticulartrueifpN =αiskeptfixed as N . →∞ For these definitions to make sense, one has to show that the number of isolated points and the core are well 3 Leaf removal process and the defined, that is, do not depend on the choice of history. The formal proof is given in the appendix. core of a graph Global leaf removal process: Start from a graphG. Our aim is to define, for any (random or not) finite graph Ifithasnoleaves,stop. Elseselectoneleafineverybunch G, a remarkable subgraph which we call the core of G. It of leaves. Remove from the vertex set V both extremities is obtained by leaf removal, an operation that we define of all the selected leaves, and define V(1) to be the set now. of remaining vertices. Let G(1) be the subgraph of G in- Leaf: A leaf of a graph G is a couple of vertices (v,w) duced by V(1). Note that the leaves of G(1) (if any) are such that v,w is an edge of G and w belongs to no not leaves of G. In this operation, the vertices belonging { } other edge of G. Note that this is not the most standard to a bunch of G that were not selected become isolated definitionandthat(v,w)and(w,v)arebothleavesifand points of G(1). They remain isolated for the rest of the only if v,w is a connected component of G. process. Iterate the procedure and define G(2),G(3), { } ··· Bunch of leaves: Abunchofleavesisamaximalfam- until a graph without leaves is obtained. ily of leaves with the same first vertex. The leaves of a Inthe proofgivenin the appendix thatthe coreiswell- graphcan be grouped into bunches of leaves in an unique defined,theargumentinstep3cimpliesinparticularthat way. leafremovalsthattakeplaceindistinctbunchesofagraph Leaf removal: If (v,w) is a leaf of G, and G′ the sub- commute. It implies that the global leaf removal process graph of G induced by V v,w , we say that G′ is ob- leads to same core and number of isolated points as any \{ } tained from G by leaf removal of (v,w). In other words, step by step leaf removal process. G′ is obtained from G by removing vertices v and w, the edge v,w andallotheredgestouchingv. Notethatthis While the global leaf removal process is convenient for { } analyticalcomputations,thestepbystepleafremovalpro- operation can destroy other leaves of G and also create cess is easier to implement on the computer. new leaves. See Fig. 1 for a pictorial example. Step by step leaf removal process: Start from a graph G. If G has no leaves, stop. Else, choose a leaf (v,w) and remove it, leading to a graph G′. If G′ has no M. Bauer and O. Golinelli — Core percolation in random graphs: a critical phenomena analysis 4 4 Core percolation: infinite N re- bors of the root, and the structure of these trees involves sults lowerordercontributions. Incontributionstopn,theroot has at least one neighbor whose attached tree contributes to q and any number of neighbors contributing to q or The global leaf removal process allows to compute the n 1 q or or q . So most salient characteristics of the leaf removal process, 2 n−1 ··· the functions i(α), c(α) and l(α). Remember that by def- p =e−α(eαqn 1)eα(qn−1+···+q1). n inition, the number of isolated points after leaf removal − is Ni(α) + o(N), the number of vertices in the core is Analogously, in contributions to q , the root has at least n N =Nc(α)+o(N), and the number of edges in the core oneneighborwhoseattachedtreecontributesto p and c n−1 is L = Nl(α)+o(N). The clue is an enumeration of all anynumberofneighborsnoneofwhichcontributingtop c 0 the configurations on the random graph that contribute or p or or p . So 1 n−1 ··· extensively to the fundamental events in the global leaf removal process at step n (which goes from G(n−1) to qn =e−α(eαpn−1 1)eα(1−pn−1−···−p0). − G(n)): emergence of a new isolated point, removal of a These two relations allow it to be shown that point, removal of an edge1. This enumeration is possible because the finite configurations of vertices and edges in p = e e forn 0 n 2n+1 2n−1 − ≥ therandomgraphwithextensivemultiplicityaretree-like. q = e e forn 1, n 2n−2 2n This implies that the problem has a recursive structure. − ≥ The weight of a tree is chosen so as to reproduce the cor- where e (α) is the sequence of iterated exponentials, de- n rect random graph weight: a vertex has weight e−α and fined by an edge has weight α. One has to be rather careful to avoid double counting and omissions; the examination of e−1 =0 and en =e−αen−1 forn 0. (1) ≥ all cases is very tedious so we omit the details and simply The events on the random graph that at step n of the outline the strategy. global leaf removal process a given vertex becomes iso- The key ingredient is a study of leaf removal on rooted lated, or that a given vertex disappears, or that a given trees. Startingfromarootedtree,weapplythegloballeaf edgedisappearscanallbe interpretedintermsofthe pre- removalprocess,withthe conventionthat evenif the root vious configurations. In each case, the different possible has only one neighbor, it is not counted as a standard contributions have to be taken into account, and also the leaf2. We let p , n 0 be the generating function for n ≥ rule that a root with a single neighbor can be touched by rooted trees whose root becomes isolated exactly at step leafremovalhastobe enforced. We omitthis painfulcase n of the globalleaf removalprocess, and q , n 1 be the n ≥ by case analysis and only state the results. generatingfunctionforrootedtreeswhoserootisremoved The explicit formulæ for extensive contribution to the exactlyatstepnofthegloballeafremovalprocess. Forin- averagenumberNi (α)ofisolatedvertices,Nc (α)ofnon n n stance,p countsrootedtreeswithanisolatedroot,hence 0 isolated vertices and Nl (α) of edges in G(n) are trees with a single vertex, and p = e−α. As another ex- n 0 ample, q1 counts rooted trees whose root touches at least in(α) = e2n+1+e2n+αe2ne2n−1 1, − one standard leaf. Consider the trees whose root touches c (α) = e e αe e +αe2 , (2) exactly k 1 standard leaves, and l 0 other vertices. n 2n− 2n+1− 2n 2n−1 2n−1 α These l ve≥rtices can be seen as roots≥of non-trivial sub- ln(α) = (e2n e2n−1)2. 2 − trees of the original tree, so by definition they contribute to 1 p . So the weight is Now comes the crucial fact: when α e, the sequence − 0 e (α) converges to W(α)/α, where W(α≤) is the Lambert n e−α(αe−α)k (α(1−p0))l. function, defined for α ≥ 0 as the unique real solution of k! l! the equation WeW = α. The function W(α) is analytic Hence for α 0. When α>e, W(α)/α remains a fixed point of ≥ the iterationEq. (1) but it is unstable: the sequence e q1 =e−α (αe−α)k (α(1−p0))l =1 e−αe−α. is oscillating. However the even subsequence e2n and{odnd} k! l! − subsequence e are still convergent. The even limit is k≥1l≥0 2n+1 XX strictly largerthanthe oddlimit. We define the functions Inthesameway,contributionstop orq forlargern′s n n A(α) and B(α) for α 0 by canbeanalyzedintermsofthetreesattachedtotheneigh- ≥ 1 The method of Refs. [3, 4] relies on approximate differential lim e = B and lim e = A. 2n 2n+1 equationsthatapplytoaslightlydifferentmodelofrandomgraphs. n→∞ α n→∞ α It is very powerful, but less intuitive than the direct enumeration methodthatfollows. Then (A,B) solve the system 2 However, a configuration where a neighbor of the root is a standardleafistreatedasusual. AeB =α, BeA =α. (3) M. Bauer and O. Golinelli — Core percolation in random graphs: a critical phenomena analysis 5 2 negative ε, A(α)=B(α)= 1 3 19 185 2437 B 1+ ε ε2+ ε3 ε4+ ε5+O(ε6) 2 − 16 192 − 3072 61440 while for small positive ε, W 61/2 3 A(α) = 1 61/2ε1/2+2ε ε3/2 ε2+O(ε3/2), 1 B − − 20 − 5 = A 61/2 3 W = B(α) = 1+61/2ε1/2+2ε+ ε3/2 ε2+O(ε3/2). 20 − 5 For i(α), this implies that there is a jump only in the second derivative of at the transition, with A 1 ε2+O(ε3) forε<0 3 e 1 2e i(α)= − ε+ e − e  0  2eε2+O(ε3) forε>0 0 1 2 3 4 Connectivity α while c(α) and l(α) havea jump in the first derivative at the transition, with Figure 2: The special functions W(α), A(α) and B(α), 0 forε<0 which coincide for α≤e. c(α)=( 12ε 4(6)1/2ε3/2 54ε2+O(ε5/2) forε>0 e − e − 5e For α e, the unique solution is A=B =W. For α>e, and ≤ the previous solution becomes unstable and (A,B) is the 0 forε<0 solution of Eq. (3) selected by the rule A<W <B. This l(α)= 12ε 54ε2+O(ε3) forε>0. is summarized on Fig. 2. (cid:26) e − 5e Taking the limit in Eq. (2) leads to The expansion for l contains only integral powers of ε, A+B+AB and this may be related to the fact (see section 5) that i(α) = 1, α − the finite size corrections for the number of edges in the (B A)(1 A) core Lc are slightly nicer than the ones for the number of c(α) = − − , (4) vertices in the core N . The average connectivity of the α c (B A)2 core is l(α) = − . 2α 2l(α) B A 2(6)1/2 4 α = = − =2+ ε1/2+ ε+O(ε3/2) eff For α e, c(α) = l(α) = 0, and the core indeed has a c(α) 1 A 3 3 ≤ − size o(N). On the other hand, the core occupies a finite forε>0,whichimpliesthatgiantcorecomponentshould fraction c(α) of the vertices for α > e. The behavior of look like a large loopfor α close to e+. The exponent 1/2 Eq (1) is responsible for this geometric transition, core percolation,atα=e. Asc(α)andl(α) 0whenα e+, inthefirstcorrectionmakessuchapropertyquitedifficult → → to see numerically at large but finite N. these functions are continuous but their derivatives are In the random graph model, the vertices do not live in not: the transition is of second order. Note again that any ambientspace, andthe notionof correlationlength is core percolation appears at α = e, contrary to backbone ambiguous. This problem will reappear in the finite size percolation, which occurs at α=1. scaling analysis of the next section. However, the emer- Thefactthatthecoreofagraphisaninducedsubgraph genceofthecoreisveryreminiscentofcriticalphenomena ofthe originalgraphallowsto givea physicist’s argument in physics. In particular, the critical slowing down is ob- fortheuniquenessofthegiantcomponentinthecore. Fix servable during the global leaf removal process. Indeed, α > e. If the core of the random graph contains two or the speed of convergence of the iterated exponential se- morelargeconnectedcomponents,therewasnoedgewith quence can be computed. One finds that for α = e, the extremities in two components in the original graph. But 6 convergence is exponential: the convergence rate ξ−1(α) as the total size of the large connected components is of is given by the formula orderN,theabsenceofsuchanedgeisextremelyunlikely. The behavior of thermodynamic functions close to the A+B transitionis the following. Writing α=e(1+ε),for small ξ−1 = logα, 2 − M. Bauer and O. Golinelli — Core percolation in random graphs: a critical phenomena analysis 6 and more precisely As we want to simulate graphs with N up to 106 and with an average connectivity α of order O(1), we must W e ( )nwe−n/ξ forα<e, use an algorithm which requires computer memory and n − α ∼ − time of order O(N), not O(N2). With the microcanoni- A e ae−(2n+1)/ξ forα>e, cal ensemble, the program is simpler: a random graph is 2n+1 − α ∼ − obtained by choosing at random L distinct edges among B e be−2n/ξ forα>e, all the possible edges. From a Monte-Carlopoint of view, 2n − α ∼ the microcanonical ensemble has another advantage: the where a(α), w(α) and b(α) are positive functions (they measurements fluctuate less. coincide for α<e and be−B/2 =ae−A/2). When α e−, As the graph (or equivalently its adjacency matrix) is ξ 2e , and when α e+, ξ e . → very sparse, it is stored in an array T of 2L integers, in- ∼Ate−αα=e, the conver→gence is∼algαe−beraic, and dexed by an arrayK of N +1 integers; the set of vertices adjacent to the vertex v is the arraysection T(i) where 1 K(v) i < K(v + 1). Then the connect{ivity}of v is e en = ≤ − e K(v+1) K(v). This defines the arrayK,withthe rules (cid:18) (cid:19) − 61/2 3 21(6)1/2logn 1 K(1)=1 and K(N +1)=2L+1. ( )n + +( )n+1 +O( ). Note that each edge v,w appears twice in T: once − n1/2 2n − 80 n3/2 n3/2 { } for v and once for w. This requires twice more memory This leads to the asymptotics at α=e: than a storage method exploiting the symmetry of the matrix (in which the edges appear only once), but the 3 e 1 i = − +O , computational task is faster because the adjacent vertices n e n3/2 (cid:18) (cid:19) of a given vertex are simply obtained from arrays T and 6 31/2 21logn 1 K. c = 1 +O , n en − 4n1/2 − 80 n n The leafremovalprocessisdone leafbyleaf. Eachtime (cid:18) (cid:18) (cid:19)(cid:19) 6 21logn 1 a leaf is removed, the adjacent vertices are examined: if ln = 1 +O . new leaves appear, there are added to a list of potential en − 80 n n (cid:18) (cid:18) (cid:19)(cid:19) leaves to be considered later. Each elementary leaf re- The first correctionfor c is more important that the one moval requires a computational time of order O(1) and n for l . Moreover the logarithms at α = e lead to suspect not O(N). So the computational time for the global leaf n that the finite size analysis of the next section might also removal is proportional to the number of removed leaves, be complicated by logarithms. which is bounded by N/2. Then the total computational timeforonerandomgraph(generationandleafremoving) is N times a function of α. 5 Numerical studies of the core For each random graph, we have measured the num- percolation ber of isolated points I , the size (number of vertices) | | of the core N , the number of edges of the core L and c c The analyticalcomputations above haveenabled us to lo- consequentlythe averageconnectivity ofthe core2L /N : c c cate a phase transition at α = e. They give information there are estimators of Ni(α), Nc(α), Nl(α) and α , eff concerning the critical region but do not exhaust all the as defined before. As usual in Monte-Carlo simulations, critical exponents. So we made an extensive numerical averages have been done over many random graphs, and analysis of the core using Monte-Carlo simulations. At their confidence intervals (or error bars) are estimated by the first step this can also be used to check the previous the variance of the measurements. analytical results. But let us start with the numerical al- For eachvalue ofα, wehavegenerated10000graphsof gorithm. sizeN =100,N =1000andN =10000,and1000graphs of size N =100000 and N =1000000. At the transition 5.1 Monte-Carlo algorithm value α = e, 10000 graphs have been generated for all the sizes. The whole computation takes a few days on a Our Monte-Carlosimulations consist in generating lots of medium Sun workstationwithout special optimization. random graphs, removing leaves step by step, and study- ing the remaining cores and isolated points. More pre- 5.2 Monte-Carlo results cisely, for a given set of parameters (N,α), we generated random graphs in the microcanonical ensemble, with N On Fig. 3, Monte-Carlo averages of N /N, L /N and c c verticesandL=Nα/2edges(Lisroundedtothenearest 2L /N are compared with the infinite N results, c(α), c c integer value). In the microcanonical ensemble the total l(α) and α . Errors bars are not drawn because they eff number L of edges is fixed (in contrast to the canonical are smaller than the size of symbols. This figure is a typ- ensemble in which L fluctuates). ical case of a second order transition. Far from the tran- M. Bauer and O. Golinelli — Core percolation in random graphs: a critical phenomena analysis 7 2.5 2.7 2.9 3.1 3.3 150 2.0 2.5 3.0 3.5 4.0 0.8 Variance of the size Size of the core: N /N c 0.6 100 N = 1000 N = 10000 N = 100000 N = 1000000 0.4 50 N = 1000 0.2 N = 10000 N = 100000 N = 1∞000000 0 N = 0.0 Variance of the edges Edges of the core: L /N c 150 1.0 100 50 0.5 0 2.5 2.7 2.9 3.1 3.3 0.0 Connectivity α Connectivity of the core: 2L /N Figure 4: Monte-Carlo estimations of the variance of the c c size of the core χ (α) and the variance of the number of c 3.5 edges of the core χ (α). l sition, differences between finite N and thermodynamic 3.0 (i.e. N = )functionsaresmall. Finitesizeeffectsareat ∞ least of order 1/N, because the analytical calculations do nottakeintoaccounttheloopsoffinitesize: theirnumber 2.5 is O(1), so their contributions are O(1/N). The simplest example is the “triangle”subgraphmade of three vertices and three edges, not connected to the rest of the graph. Obviously, the triangles have no leaf and belong to the 2.0 core: their contribution to N is (αe−α)3/2. c 2.0 2.5 3.0 3.5 4.0 We have verified that finite size effects are of order Connectivity α O(1/N) (but not larger) for c(α) and l(α) far from the transition. For α , this is probably true but less clear eff Figure 3: Monte-Carlo averages (symbols) and analytical because fluctuations are stronger. On the other hand, in results (solid line) for the size, edges and average connec- the critical region α e, finite size effects are larger and ∼ tivity of the core. somecriticalexponentscanbedefined. Theyarediscussed later. M. Bauer and O. Golinelli — Core percolation in random graphs: a critical phenomena analysis 8 3.5 0 1 2 3 4 1.0 Variance of the connectivity 3.0 Isolated points: |I|/N 2.5 χ)α 2.0 g( 0.5 l 1.5 N = 1000 N = 1000 N = 10000 1.0 N = 10000 N = 100000 N = 100000 N = 1000000 N = 1000000 ∞ N = 0.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 0.0 lg(α−e) Variance Figure 5: Monte-Carlo estimations of χ (α) (variance of α the average connectivity of the core) versus (α e). The − 0.10 axes are labeled by decimal logarithms. Wehavealsoexaminedthevariancesofthesize,number of edges and average connectivity of the core. For α not tooclosetoeandlargeN,weexpectthatthefluctuations 0.05 (square root of the variance) are of order O(√N) for N c and L , and O(1/√N) for α . So to obtain a large N c eff limit, we define χ (α) Var(N )/N, χ (α) Var(L )/N c c l c ≡ ≡ and χ (α) N Var(2L /N ). These quantities are anal- α c c ≡ ogous in the spin models to the magnetic susceptibility 0.00 0 1 2 3 4 (equivalent to the fluctuations of the magnetization). Connectivity α On Fig. 4, Monte-Carlo estimations of χ and χ show c l that χ (α) and χ (α) have a finite limit for N = when c l ∞ α > e, a vanishing limit when α < e and diverge when α Figure6: Monte-Carlo averages I /N and variance χ (α) i | | approaches the critical value e. By analogy with c(α) of the number of isolated points. The solid line is the an- ∼ l(α) (α e) and α 2 (α e)1/2 when α e+, alytical result i(α) for N = . ∼ − eff − ∼ − → ∞ powerlawsareexpectedforthedivergences. So,wedefine two critical exponents ρ and ρ′ by are so smallthat they are not visible. On the other hand, χc(α)∼χl(α) ∼ (α−e)−ρ, (5) the variance χi(α) = Var(|I|)/N shows bigger statistical χ (α) (α e)−ρ′, (6) fluctuations, but the finite size effects remain small. This α ∼ − variance does not diverge anywhere. However we see a whenα e+. The exponentρcouldbe numericallymea- cusp when α e+ compatible with → → suredbyplottinglog(χ )orlog(χ )versuslog(α e). Un- c l − fortunately, this gives poor results because the curvature χ (e) χ (α) (α e)τ i i − ∼ − ofthe plotis too important,with a slopeρ changingfrom 1to0.5. Butitispossibleforα . Fig.5isalog-logplot with estimations χ (e) = 0.095(5) and τ = 0.6(1). As eff i of χ (α) versus (α e). Symbols are lined up correctly τ <1, the first derivative is infinite at α=e+. α − and the global slope gives the estimation ρ′ =1.5(1). 5.3 Finite size scaling The studies of isolated points are resumed on Fig. 6. WenowconcentrateonthefiniteN behavior,firstexactly Monte-Carlo averages of I /N are compared with the in- at the transition α = e and then in the critical region | | finite N results, i(α): errors bars and finite size effects around this transition. M. Bauer and O. Golinelli — Core percolation in random graphs: a critical phenomena analysis 9 Finite size effects at the transition −0.5 1.0 −1.0 0.5 −1.5 2L/N−2 c c σ (2L/N) c c L/N σc (L/N) Scaling distributions at the transition c N/N −2.0 σ c(N/N) Size: Prob(N /Nω ≤ x) c c 0.0 2 3 4 5 6 0 1 2 3 4 5 6 lg(N) 1.0 Figure 7: Top to bottom: the average connectivity (mean and width), the number of edges (mean and width) and the size (mean and width) of the core versus the size N of the random graph. The axes are labeled by decimal logarithms. The negative slopes are measurements of φ (for the connectivity) and ω 1 (for size and edges). − 0.5 − By analogy with the classical percolation transition at α = 1 where the size of the largest connected component is [1] of order O(N2/3) and its average connectivity Edges: Prob(L /Nω ≤ x) is 2+O(1/N2/3), we postulate the existence ofother crit- c ical exponents ω and φ defined by 0.0 0 1 2 3 4 5 6 7 8 N L Nω, (7) 1.0 c c ∼ ∼ α 2 N−φ (8) eff − ∼ when N at α = e. This hypothesis is tested on → ∞ Fig. 7: data are correctly fitted by power-laws with N = 100 N = 1000 ω =0.63(1) and φ=0.21(1). N = 10000 0.5 N = 100000 Ofcourse,ifthe largeN behavioris modifiedby a(power N = 1000000 ofa)logarithmicfunction,thetruevaluesoftheexponents are different than their apparent values when N is large butfinite. Heretheseexponentsaredeterminedbyconsid- eringtheaveragesoftheMonte-Carlomeasurements. The Connectivity: Prob((2L /N −2)Nφ ≤ x) width σ of their distributions are also plotted on Fig. 7: c c means and widths have similar slopes. Consequently 0.0 0 1 2 3 4 χ (e) χ (e) N2ω−1 and χ (e) N1−2φ (9) c ∼ l ∼ α ∼ Figure8: Cumulativedistributionfunctionsofthesize,the number of edges and the average connectivity of the core, diverge when N at α=e. →∞ as functions of their respective scaled variables. We have AswidthsandmeansoftheMonte-Carlomeasurements set ω =0.63 and φ=0.21. are of the same order, the distributions remain broad in the large N limit at the transition. On the contrary, when α = e, the distributions are sharp. On Fig. 8, 6 M. Bauer and O. Golinelli — Core percolation in random graphs: a critical phenomena analysis 10 the cumulative distribution functions Prob(N /Nω x), c ≤ Prob(L /Nω x) and Prob((2L /N 2)Nφ x) are Finite size scaling in the critical region c c c plotted as fun≤ctions of the scaling vari−able x fo≤r α = e. 1.5 We observe that the curves converge when N is large to scalingdistributions(independentofN)andthisconfirms θ = 0.37 the hypotheses Eq. (7) and Eq. (8). α)) 1.0 tioWnshednecxrebaseecolmikeesalarGgaeu,stshieasne.scCaloinnsgeqduisetnritblyu,titohneyfuanrce- N − l( NN == 110000000 not large distributions in the sense that every moment is L / c N = 100000 finite, inagreementwithEq.(9). Onthe otherside,these θN ( 0.5 N = 1000000 functions seem to be power laws for small x. This allows to define another exponent δ with Prob(N /Nω x) Prob(L /Nω x) xδ (10) c c ≤ ∼ ≤ ∼ 0.0 when x 0. Our estimation is → −10 −5 0 5 10 δ =0.36(3). (α − e) Nθ The numericalvalues suggestthat ω+δ =1, but we have Figure 9: Finite size scaling for the edges of the core in no argument to explain it. the critical region. Byconsideringtheprobabilitythatthecoreofarandom graph is void at α=e, we measured a new exponent where the scaling function Q˜(y) is defined by η =0.25(1) Q˜(y) lim Nγθ Q(N, e+y/Nθ), where η is defined by ≡N→∞ which behaves as Prob(N =0) N−η. c ∼ Q˜(y)y→+∞yγ. ∼ The limit x 0 in Eq. (10) gives the conjectured rela- → As y =0 exactly at the transition α=e, tion η = ωδ, which is numerically acceptable. With the hypothesis ω+δ =1, it gives η =ω(1 ω). Q(N,e) N−γθ. − WehavealsoconsideredProb(L =N ),i.e. the proba- ∼ c c bilitythattheaverageconnectivityofthecoreisexactly2. So the exponent of finite size effects at the transition and Inthiscase,thecoreismadeofoneorseveralsimpleloops theexponentofcriticalbehaviorforN = inthevicinity ∞ withoutbranching. TheMonte-Carlostudyindicatesthat of the transition are related by θ. This remark is useful the large N limit could be a pure number: 0.12(2). More only if different quantities share the same θ. For usual intensivesimulationsareneededtoconfirm(orinvalidate) models of statistical physics with a 2-D or 3-D geometry this result. (like classical spin systems), the exponent θ describes the More relations between critical exponents can be ob- divergence of the correlation length ξ. So the uniqueness tained by using the finite size scaling hypothesis [9]: in of ξ implies the uniqueness of θ. Unfortunately for ran- the vicinity of the transition, the behavior of finite random graphs, we have no equivalent length and no simple dom graphs is determined by the scaling variable phenomenological interpretation for θ. However we shall assume that θ is unique. y =(α e)Nθ As wehavecomputed exactN = formulæ in Sect.4, − ∞ we can directly study Q(N,α) Q(α), i.e. the finite size where θ is a positive scaling exponent. effects. The scalingfunctionisn−owQ˜(y) yγ andis max- Firstwe shortly resume the scaling theory for a general imal around y = 0. From a numerical p−oint of view, the quantityQ(N,α),forsizeN andconnectivityα. ForN = analysis becomes easier than the one of the monotonic , let us suppose that function Q˜(y). ∞ LetusnowconsiderN /N andL /N. Asc(α) l(α) Q(α) (α e)γ c c ∼ ∼ ∼ − (α e) when α e+, for these quantities γ =1. Then − → when α e+ (γ could be positive or negative). Then we θ =1 ω. expect in→the critical region that − On Fig. 9, with the choice θ = 0.37 (induced by the nu- Q(N,α) N−γθ Q˜(y) merical measure of ω), Nθ(L /N l(α)) is plotted versus c ∼ −