Isomorphism and Embedding Problems for Infinite Limits of Scale-Free Graphs

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1 Isomorphism an Embeing Problems for Infinite Limits of Scale-Free Graphs Robert D. Kleinberg Jon M. Kleinberg Abstract The stuy of ranom graphs has traitionally been ominate by the closely-relate moels G(n, m), in which a graph is sample from the uniform istribution on graphs with n vertices an m eges, an G(n, p), in which each of the ( n 2) eges is sample inepenently with probability p. Recently, however, there has been consierable interest in alternate ranom graph moels esigne to more closely approximate the properties of complex real-worl networks such as the Web graph, the Internet, an large social networks. Two of the most well-stuie of these are the closely relate preferential attachment an copying moels, in which vertices arrive one-by-one in sequence an attach at ranom in rich-get-richer fashion to earlier vertices. Here we stuy the infinite limits of the preferential attachment process namely, the asymptotic behavior of finite graphs prouce by preferential attachment (briefly, PA graphs), as well as the infinite graphs obtaine by continuing the process inefinitely. We are guie in part by a striking result of Erős an Rényi on countable graphs prouce by the infinite analogue of the G(n, p) moel, showing that any two graphs prouce by this moel are isomorphic with probability 1; it is natural to ask whether a comparable result hols for the preferential attachment process. We fin, somewhat surprisingly, that the answer epens critically on the out-egree of the moel. For = 1 an = 2, there exist infinite graphs R such that a ranom graph generate accoring to the infinite preferential attachment process is isomorphic to R with probability 1. For 3, on the other han, two ifferent samples generate from the infinite preferential attachment process are non-isomorphic with positive probability. The main technical ingreients unerlying this result have funamental implications for the Department of Mathematics, MIT, Cambrige MA rk@math.mit.eu. Supporte by a Fannie an John Hertz Founation Fellowship. Department of Computer Science, Cornell University, Ithaca NY kleinber@cs.cornell.eu. Supporte in part by a Davi an Lucile Packar Founation Fellowship an NSF grants an structure of finite PA graphs; in particular, we give a characterization of the graphs H for which the expecte number of subgraph embeings of H in an n-noe PA graph remains boune as n goes to infinity. 1 Introuction For ecaes, the stuy of ranom graphs has been ominate by the closely-relate moels G(n, m), in which a graph is sample from the uniform istribution on graphs with n vertices an m eges, an G(n, p), in which each of the ( n 2) eges is sample inepenently with probability p. The first was introuce by Erős an Rényi in [16], the secon by Gilbert in [19]. While these ranom graphs have remaine a central object of stuy an continue to have many important applications in combinatorics an theoretical computer science, recently there has also been a great eal of interest in alternative ranom graph moels whose properties more closely resemble those of complex real-worl networks such as the Web graph, the Internet, an large social networks. Two of the most well-stuie of these are the closely relate preferential attachment an copying moels; the former was introuce by Barabási an Albert in [3] an subsequently formalize by Bollobás an Rioran in [8], while the latter was introuce by Kumar et al. in [22]. A ranom graph in the preferential attachment moel (henceforth, the PA moel) is built up one vertex at a time, with each new vertex v linking to the preceing ones by new eges, where the out-egree is a parameter of the moel. Roughly, the hea of each ege emanating from v is chosen by sampling from the preceing vertices with probabilities weighte accoring to their total egree (in-egree plus out-egree); this is the preferential, or rich-get-richer, aspect of the moel, since noes of higher in-egree attract new incoming eges more reaily. (We will sometimes use the term PA graph as an informal shorthan to refer to a ranom graph rawn from the istribution efine by the PA moel.) As we iscuss further below, there has been consierable work aime at etermining funamental graph-theoretic properties in the PA moel, exposing both similarities an contrasts

2 with the classical G(n, p) moel. In the present paper, we seek to unerstan the infinite limits of the PA moel namely, the asymptotic behavior of graphs prouce by this moel as the number of noes goes to infinity, an the istribution PA on ranom graphs with countably many vertices obtaine by continuing the PA process inefinitely. We were inspire by the following classical theorem about the infinite version of the G(n, p) moel [17]. Theorem 1.1. Let G(, p) enote the probability istribution on graphs with vertex set N, in which each ege (i, j) is inclue inepenently with probability p. (Here p is any constant in (0, 1).) There exists an infinite graph R, such that a ranom sample from G(, p) is isomorphic to R with probability 1. When one first encounters this theorem, it can seem quite startling: infinite ranom graphs are not ranom at all; they are almost surely isomorphic to a single fixe graph R. A rich theory has evelope aroun the infinite moel G(, p), with connections reaching into mathematical logic, algebra, an a number of other areas (see e.g. [13]). On the other han, essentially nothing is known about the the infinite version of the PA moel. Does something analogous to Theorem 1.1 hol here as well, or is the situation funamentally ifferent? At a more fine-graine level, we are also intereste in unerstaning what can be sai about the local structure of finite graphs prouce by the PA moel as the number of noes goes to infinity. As we iscuss further below, the only prior work aressing the infinite graphs generate by such processes, as far as we are aware, are some interesting recent papers by Bonato an Janssen [11, 12], which propose the notion of stuying infinite limits of ranom graph evolution processes relate to the copying moel of [23]. These papers consier the relationship between such infinite ranom graphs an certain eterministic ajacency axioms. Some of these axioms have a unique infinite moel up to isomorphism, while others are satisfie with probability 1 by the infinite limits of the ranom graph processes consiere in these papers. However, none of their theorems resolve the question of whether an analogue of Theorem 1.1 hols for such infinite ranom graphs. Our first result is the following, where again PA enotes the istribution associate with the infinite PA moel. Theorem 1.2. For = 1, 2, there is a graph R such that a ranom sample from PA is isomorphic to R with probability 1. For = 1 this is clear, since the outcome of the ranom process will almost surely be a tree with countably many noes, in which each noe has infinite egree. For the case of out-egree = 2, the resulting graph R2 is much more complicate. Its structure can be characterize axiomatically, but it is also possible to give explicit constructions of graphs isomorphic to R 2. For example, it is isomorphic to the graph whose vertices consist of all finite roote binary trees with integer labels, where the vertex corresponing to a labele tree T has eges to its left sub-tree an to its right sub-tree. The global structure of the proof for the case = 2 is a stanar back-an-forth argument, which will be familiar to reaers acquainte with Theorem 1.1. The key step, however establishing that there is an aequate supply of vertices to sustain the back-anforth construction of the isomorphism is much more complicate than in the classical case, since the PA process introuces ifficult conitioning problems. One might imagine that for the cases of out-egrees = 3, 4, 5,... one coul establish isomorphisms with probability 1 to increasingly complex graphs R 3, R 4, an so on. But in fact, we have the following result. Theorem 1.3. For each out-egree 3, it is not the case that two inepenent ranom samples from PA are isomorphic with probability 1. This contrast between the cases of = 2 an 3 comes to us as something of a surprise, since it oes not have an obvious analogue in the prior work on graphs generate accoring to the PA process. There, typically, the out-egree has a clear quantitative effect on the unerlying graph parameters, but not a qualitative effect of this sort. This contrasting pair of results is a particularly succinct consequence of one of the main technical components of the paper, which aresses a funamental structural issue for both the finite an infinite versions of the PA moel a characterization of the graphs H for which the expecte number of subgraph embeings of H in an n-noe PA graph remains boune as n goes to infinity. Phrase equivalently as a statement about the infinite moel PA, we show that if a finite graph H is equal to its 3-core (i.e. the union of all subgraphs of H of minimum egree 3), then the number of subgraph embeings of H in a ranom sample from PA has a positive finite expectation, while if H is not equal to its 3-core, then the number of embeings is almost surely either zero or infinite. The existence an relative abunance of small subgraphs is a topic of consierable interest for both empirical stuies of real networks an for theoretical stuies of their moels (see e.g. [20, 23]). Our characterization theorem has a natural interpretation in this context, as

3 a precise statement about the lack of ense local structure in PA graphs G. First, any graph H of minimum egree 3 appears a boune number of times in expectation as a subgraph of G, inepenent of the size of G. Secon, any graph H = (V, E) for which E / V > 2 has a non-trivial 3-core, an so our result implies that in any PA graph G, there exists a set of noes S in G of boune expecte size, such that any embee copy of H in G inclues at least one noe from S. (In other wors, S serves as a boune set of attachment points for copies of H.) This characterization theorem for subgraph embeings yiels the non-isomorphism theorem for PA with 3 fairly irectly; it also has the following further consequence for finite PA graphs. (Here the istribution on n-vertex graphs prouce by the PA process will be enote by PA (n).) Theorem 1.4. For 3, there exist first-orer graph properties which o not satisfy a zero-one law for PA (n), i.e. there is a first-orer formula φ(g) such that 0 < lim n Pr G PA (n) (φ(g)) < 1. This contrasts with the situation for G(n, p), where it is known that every first-orer formula satisfies a zero-one law. (For a very interesting an eep analysis of firstorer properties of G(n, p) when p is a function of n, we refer the reaer to [28].) Finally, it is worth briefly returning to the original motivation for these types of moels the complex structures of graphs such as the Web, the Internet, an large social networks. The (finite) PA an copying moels are of course stylize abstractions esigne to capture some of the observe properties of these networks; they were not intene as faithful representations of the complexities of the true structures. Our stuy of infinite analogues here follows a theme that is common in a number of areas, to try gaining insight into extremely large finite systems by moeling them as infinite as, for example, when working with infinite lattice structures in physics, or with a continuum of agents in economics. Thus far, asie from the work of Bonato an Janssen [11, 12], this has not really been attempte for complex networks, but the results about finite structures that emerge from the stuy of infinite limits of the graph generation process here provie a suggestion for the kins of results one can obtain from this style of investigation, an we feel there is clearly room for further stuy in this irection. 1.1 Relation to prior work The preferentialattachment moel of ranom graphs was introuce by Barabási an Albert in [3], motivate in part by the goal of explaining the power-law egree istribution observe in the Internet topology by Faloutsos et al [18] an in the Web topology by Kumar et al [21]. Barabási an Albert s original paper containe a heuristic argument establishing a power law for the egree istribution of ranom preferential-attachment graphs; rigorous mathematical proofs of this result subsequently appeare in [1, 10]. An alternative ranom graph moel with powerlaw egree istribution, the evolving copying moel, was inepenently propose an analyze by Kumar et al [22, 23], with the aim of moeling the Web graph. Cooper an Frieze introuce a moel which simultaneously generalizes these two ranom graph moels, an again prove that the egree istribution obeys a power law [14]. A irecte version of the preferentialattachment moel was introuce an stuie by Bollobas et al in [6], who again establishe a power-law istribution both for the in-egrees an the out-egrees. In aition to their egree istribution, many other properties of preferential-attachment ranom graphs have been rigorously analyze; these inclue their iameter [8], conuctance [25], eigenvalues [24, 15], clustering coefficient [7], an robustness uner ranom vertex eletions [9]. See [2, 7, 26] for various surveys of work in this area, focusing on ifferent research communities. As iscusse above, the only other work to our knowlege that aresses the infinite graphs which arise as the limit of such processes is [11, 12]. In [11], Bonato an Janssen formulate a copying moel, similar to that propose in [23], an they show that an infinite ranom graph generate accoring to this process satisfies a certain eterministic ajacency property which they label Property (B). They then stuy various moeltheoretic an combinatorial properties of graphs satisfying property (B) an its generalizations. Of particular relevance, for our purposes, is their theorem that there are 2 ℵ0 many non-isomorphic graphs satisfying property (B). While this suggests the possibility that ranom samples from their copying moel are not almost surely isomorphic, the authors explicitly refrain from aressing this question since their focus is on stuying infinite graphs satisfying the eterministic property (B) an its generalizations, regarless of whether such graphs were generate by a ranom process or not. The subsequent paper [12], written inepenently an concurrently with our work, generalizes the ranom graph process introuce in [11] an relates it to some new ajacency properties (ARO, near-aro, local near-aro, n-near-aro). Only the ARO property has a unique infinite moel up to isomorphism; in fact, the other properties are shown to be satisfie by 2 ℵ0 many

4 non-isomorphic graphs. Moreover, the infinite ranom graphs consiere in [12] have a positive probability of failing to satisfy the near-aro property. Again, this suggests the possibility that ranom samples from this generalize copying moel are not almost surely isomorphic, but again the authors refrain from answering this question, as they leave open the possibility that the infinite graphs generate by their ranom process are almost surely isomorphic to a single infinite graph which fails to satisfy the near-aro property. 2 Definitions We begin by efining, for each > 0, a ranom graph process PA on graphs with vertex set {0, 1,...}. PA is a probability istribution on sequences of connecte unirecte graphs, G 0 G 1..., where G t has vertex set {0, 1,..., t}. Our efinition is closely moele on the efinition of the graph process (G t m ) t 0 in [8]; however, it iffers in some technical etails because we want our graphs to be connecte an theirs are potentially isconnecte. Graphs in their moel are allowe to have self-loops, an a new connecte component is create every time a new vertex appears an connects to no vertices other than itself. Our graphs will have parallel eges but no self-loops, an they will be connecte. The graph process PA is efine recursively as follows. G 0 has one vertex (labele 0) an no eges. G t+1 is obtaine from G t by aing a new vertex (labele t+1) an joining it to vertices 0, 1,..., t with ranom eges, sample inepenently at ranom from a probability istribution (the preferential attachment istribution) specifie as follows: Pr(e = (t + 1, s)) = t (s)/2t, where t (s) enotes the egree of vertex s in G t. In other wors, each neighbor of t + 1 is chosen accoring to a istribution which weights vertices by their current egree. The efinition of the preferential attachment istribution makes no sense in the case t = 0, since G 0 has no eges. Accoringly, we stipulate that vertex 1 always links to vertex 0 with parallel eges. Given a sample G 0 G 1... from PA, let G = t=0 G t an efine PA to be the resulting probability istribution on graphs with vertex set {0, 1, 2,...}. The eges of G may be numbere 1, 2,..., such that the eges of G t+1 \ G t are labele t + 1, t + 2,..., t +. An equivalent way of specifying the graph process PA woul have been to say that ege t + j (1 j ) chooses an ege uniformly at ranom from the set {1, 2,..., t}, chooses an enpoint of this ege uniformly at ranom, an joins vertex t+1 to the chosen enpoint. Although PA was efine as a probability istribution on unirecte graphs, the eges of these graphs come equippe with a natural orientation, irecte from the higher-numbere enpoint to the lower-numbere one. We will sometimes consier the graphs G as irecte graphs, an it will be clear when we are oing so. The avantage of aopting this ual viewpoint on the graphs G is that it enables us to state stronger theorems: our isomorphism theorem hols for irecte graphs sample from PA 2 an trivially implies the corresponing result for unirecte graphs; while our nonisomorphism theorem hols for unirecte graphs an trivially implies the corresponing result for irecte graphs. 3 Growth rate of vertex egrees The proofs in this paper hinge on a etaile unerstaning of the growth rate of vertex egrees, i.e. the asymptotics of t (i) as a function of t, in a typical sequence G 0 G 1... sample from PA. It has been known since the introuction of the Barabási-Albert moel that E[ t (i)] = θ( t) for any fixe i. A nonrigorous argument using ifferential equations appears in [3], an a rigorous proof may be foun in [8]. A key ingreient in our proof of Theorem 1.2 is the following stronger fact: Proposition 3.1. For any fixe vertex i, with probability 1, lim t t (i)/ t exists an is positive. Although the calculations arising in the proof are very similar to those use in establishing the asymptotics of E[ t (i)] [8], we require two more techniques from martingale theory to establish a stronger result: the existence an positivity of the limit lim t t (i)/ t. The existence of the limit is establishe using Doob s martingale convergence theorem, an its positivity comes from the Kolmogorov-Doob inequality combine with a secon-moment computation. (See [5], Chapter 35, for an introuction to martingales incluing both of the aforementione tools.) The calculations arising in these proofs are very similar to those use in Lemma 2 of [8], in which the authors prove (among other things) that E[ t (i)] = θ( t). Fix a vertex i, an consier how its egree changes at time t+1. Each of the new eges attaches to i with probability t (i)/2t, so E( t+1 (i) t (i)) = t (i)+ ( ) ( t (i) = ) t (i). 2t 2t It follows that the sequence of ranom variables ( t (i)) t i may be transforme into a martingale by

5 rescaling, as follows. Define t 1 c t = j=1 ( ) 2j X t = t (i)/c t. Now the sequence (X t ) t i is a martingale (aapte to the σ-fiel F t generate by the ranom variable G t ) since: E[X t+1 F t ] = = 1 E[ t+1 (i) F t ] c t+1 ( ) t (i) 2t c t+1 = 1 c t t (i) = X t. The constant c t is θ( t), as may be seen easily by taking the logarithm of both sies of the formula efining c t, an using the ientity x 1 2 x2 < log(1 + x) < x. The following two theorems are instrumental in the proof of Proposition 3.1. Proofs may be foun in [5], or in most books on stochastic processes. Theorem 3.1. (Doob s Martingale Convergence Theorem) Let X 1, X 2,... be a submartingale. If K = sup n E( X n ) <, then X n X with probability 1, where X is a ranom variable satisfying E [ X ] K. Theorem 3.2. (Kolmogorov-Doob inequality) If X 1,..., X n is a submartingale, then for α > 0, [ ] Pr max X i α 1 i n α E [ X n ]. Proof of Proposition 3.1. The ranom variables X t are non-negative, so E( X t ) = E(X t ) = E(X i ) = /c i for all t. This establishes that the X t satisfy the hypotheses of Doob s Martingale Convergence Theorem, so with probability 1 they approach a finite limit as t. Given that c t = θ( t), this implies that lim t t (i)/ t exists almost surely. It remains to show that the limit is almost surely positive. The iea of the proof is simple, an conceptually similar to Zeno s Paraox of the Race Course [4]. We will show that after the egree of i excees some threshol, the value of X t is very unlikely to rop by a factor of 2 from its current value. In orer for lim t X t to be zero, it must be the case that X t ecreases by a factor of two infinitely often, an event having probability 0. then you that you To make this notion precise, efine a sequence of times n 0 < n 1 <... as follows. Let n 0 = i. Let n 1 be the smallest value of n such that X n < (1/2)X n0, or if no such n exists. Continue efining n 2, n 3,... in the same manner, i.e. n j+1 is the smallest n such that X n < (1/2)X nj, or if no such n exists or if n j =. We will prove that Pr(all n j are finite) = 0, an to o so it is sufficient to prove that Pr(n j+1 < n j ) < 1 δ for some constant δ > 0. To o so, we use the Kolmogorov-Doob inequality applie to the submartingale X n = (X nj X n ) 2 (n n j ). (Any convex function applie to a martingale yiels a submartingale, by Jensen s inequality.) An estimate for E( X n X nj ) is compute in Supplementary Section A. The result is: E( X n X nj ) < (C/ n j )X nj. for some constant C. Now, by the Kolmogorov-Doob inequality, Pr(max n n j Xn > (X nj /2) 2 X nj ) = lim Pr( max N n j n N = 4X 2 n j lim E( X N X nj ) N 4Xn 2 j (C/ n j )X nj 4C nj X nj C / nj X n > (X nj /2) 2 X nj ) for some constant C. Recall that our goal is to show that Pr(n j+1 = n j ) > δ or, equivalently, that Pr(max n nj Xn (X nj /2) 2 n j ) > δ for some constant δ > 0. We now see that this coul be accomplishe by establishing that C / nj 1 δ. So to finish, it suffices to prove that nj grows unbounely large as j. (In fact, it woul suffice to prove that nj is eventually greater than C /(1 δ).) But this is easy: the probability that n = Y for all n > N is boune above by ( n=n 1 Y 2n) = 0, so with probability 1, n as n. 4 An isomorphism theorem for = 1, 2 We begin with the easy proof of Theorem 1.2 in the case = 1. Let R1 enote a countable roote arborescence in which each non-root vertex has infinite inegree an has a path to the root. Theorem 4.1. A ranom sample from PA 1 is almost surely isomorphic to R1 as a irecte graph. Proof. Let G be a ranom sample from PA 1. By construction, every vertex except for 0 has outegree 1,

6 an vertex 0 has outegree 0. With probability 1, the inegree of each vertex is infinite, by Proposition 3.1. By construction, each vertex except for 0 has a path to vertex 0. These properties uniquely etermine the isomorphism type of G as a irecte graph. For the rest of this section, we focus on the case = 2. Consier the following three axioms for an infinite irecte graph K with countable vertex set. 1. There exists a vertex v 0 with outegree 0. Every other vertex has outegree For any pair of (not necessarily istinct) vertices v, w, there are infinitely many vertices whose two outgoing eges link to v an w. 3. K oes not contain any infinite forwar path. Proposition 4.1. Any two countable irecte graphs K 1, K 2 satisfying axioms (1)-(3) are isomorphic. Proof. Let v 0, v 1,... be the vertices of K 1, orere so that all of the outgoing eges from v j link to vertices in the set {v 0,..., v j 1 }. Such an orering may be constructe recursively as follows. Choose v 0 to be the vertex with outegree 0. Given v 0,..., v j 1, start from an arbitrary vertex of K 1 \ {v 0,..., v j 1 } an follow outgoing eges until a vertex v j with outegree 0 is reache; this must happen after a finite number of steps, since otherwise K 1 woul contain a cycle or an infinite forwar path. Similarly, let w 0, w 1,... be the vertices of K 2, orere so that all of the outgoing eges from w j link to vertices in the set w 0,..., w j 1. The proof now procees by a back-an-forth argument. We will construct an isomorphism φ : K 1 K 2 by first selecting φ(v 0 ), then φ 1 (w 0 ), then φ(v 1 ), then φ 1 (w 1 ), an so on a infinitum, until a one-to-one corresponence between V (K 1 ) an V (K 2 ) has been efine. The steps in which we select φ(v i ) will be calle forwar steps, those in which we select φ 1 (w i ) are reverse steps. To start, set φ(v 0 ) = w 0. The construction now procees in a series of steps, each of which starts with a one-to-one corresponence between finite subsets S 1 V (K 1 ), S 2 V (K 2 ) inucing an isomorphism between the corresponing inuce subgraphs, an extens this one-to-one corresponence to inclue a single aitional element of each vertex set, while preserving the fact that it efines an isomorphism of inuce subgraphs. The forwar steps alternate with the reverse steps. For reasons which will soon become apparent, we a an aitional claim into our inuction hypothesis: every outgoing ege from a vertex of S 1 (resp. S 2 ) joins it to another vertex of S 1 (resp. S 2 ). This is satisfie vacuously in the base case where S 1 = {v 0 }, S 2 = {0}. To perform a forwar step, take the lowestnumbere vertex v j in V (K 1 ) \ S 1. This vertex has two outgoing eges pointing to vertices v i1, v i2 V (K 1 ). We have i 1, i 2 < j, so both v i1 an v i2 belong to S 1. Now choose φ(v j ) to be any vertex w V (K 2 ) \ S 2 such that w points to φ(v i1 ) an φ(v i2 ). (Such a vertex is guarantee to exist, by Axiom 2.) It is now easy to check that the inuction hypothesis is still satisfie. By construction, φ maps the outgoing eges from v j to the outgoing eges from w. As for the incoming eges, φ is only efine at this stage as a mapping from S 1 {v j } to S 2 {w}, an neither v j nor w have any incoming eges from vertices in these sets. (This is where we neee the aitional fact that every outgoing ege from a vertex of S 1 (S 2 ) joins it to another vertex of S 1 (S 2 ). It is trivial to check that this fact remains true after extening S 1, S 2 to inclue v, i, respectively.) This completes the proof of the inuction hypothesis in the case of a forwar step. By symmetry, the inuction hypothesis is prove for reverse steps as well. Any countable irecte graph satisfying axioms (1)- (3) will be enote by R 2. Theorem 1.2 now follows from the following more precise result. Theorem 4.2. If G is sample at ranom from PA 2, then G is almost surely isomorphic to R 2. Proof Sketch. We must check that G satisfies axioms (1)-(3) almost surely. By construction, there is a single vertex with outegree 0, all other vertices have outegree 2, an no infinite forwar paths exist in G. Finally, we require the following: Proposition 4.2. Given any two (not necessarily istinct) vertices j 1, j 2 in V (G ), there are infinitely many i V (G ) whose two outgoing eges point to j 1, j 2. The proof of this relies on Proposition 3.1. Informally, that proposition guarantees the existence of constants x 1 = lim t t (j 1 )/ t x 2 = lim t t (j 2 )/ t so for sufficiently large t, the probability that vertex t links to j 1 an j 2 is approximately (x 1 t/4t)(x2 t/4t) = x1 x 2 /16t. The probability that no vertex after t 0 links to j 1 an j 2 is approximately ( t=t 0 1 x 1x 2 ) 16t = 0. Thus, almost surely, there exists a vertex i V (G )\S 2 whose two outgoing eges point to j 1, j 2. The biggest problem with making this informal argument rigorous is that, by conitioning on the values of x 1 an x 2, we change the istribution of the ranom outgoing eges from each vertex; it is no longer the

7 preferential attachment istribution, so we have no justification for our estimate of the probability that t links to j 1 an j 2. While the informal argument gives the correct intuition, the rigorous version is surprisingly intricate; for etails, see the full version of this paper. Concrete constructions for R2. Theorem 4.2 supplies an axiomatic characterization of R2, but unlike Theorem 4.1 it oes not concretely specify a graph which is isomorphic, almost surely, to ranom samples from PA 2. In this section we present two such constructions. The first construction prouces R2 as the union of a countable chain of infinite graphs R 0 R 1..., efine recursively as follows. Let R 0 consist of a vertex v 0 of outegree 0, an countably many other vertices, each with two parallel eges pointing to v 0. Given R j, construct R j+1 as follows: for each pair of (not necessarily istinct) vertices v, w V (R j ), ajoin a countable set of new vertices, each with two outgoing eges pointing to v, w. Finally, put R 2 = R j. j=0 It is routine to verify that this graph R2 satisfies the axioms (1)-(3). The secon construction efines R2 as a graph whose vertex set is a set of labele binary trees. Specifically, let Σ be a countable alphabet, an let the vertex set V (R2 ) be the set of finite roote binary trees whose eges are labele with elements of Σ. If T V (R2 ) is a tree with more than one noe, then T has two outgoing eges in R2 pointing to its left an right subtrees. Again, it is straightforwar to verify that this efinition of R2 satisfies the axioms (1)-(3). A moel-theoretic characterization of R2. Our goal in this section is to specify a precise sense in which R2 is axiomatically characterize by the conitions given in Section 4. We will exhibit a firstorer theory T, in the language of irecte graphs, such that R2 is a prime moel of T. (A moel M of a first-orer theory T is calle prime if every other moel of T contains a submoel isomorphic to M. If a countable theory has a prime moel, this moel is unique up to isomorphism.) Interestingly, T has many other countable moels which are not isomorphic to R2. (Note the close thematic links between this section an [11].) Let T enote the following set of first-orer formulas in the language of irecte graphs, consisting of one axiom Outegree an two infinite families of axioms (Acyclic n ) n 2 an (Ajacency n ) n 1 : Outegree: G has one vertex of outegree 0, an all other vertices have outegree 2. Acyclic n : G oes not contain an n-cycle. Aj n : For any pair v, w of vertices of G, there are at least n istinct vertices whose two outgoing eges point to v, w. Proposition 4.3. R 2 is a prime moel of T. Proof. Clearly R2 is a moel of T. Given any other moel G, an isomorphic embeing φ : R2 G is constructe in a manner similar to the back-an-forth argument use in proving Theorem 4.2, except that this time the construction is one-irectional since we are not trying to make φ surjective. Let {v 0, v 1,...} be the vertex set of R2, numbere so that the eges from v j point to elements of {v 0,..., v j 1 } as before. We construct φ inuctively, by specifying that φ(v 0 ) is the unique vertex of G having outegree zero; an that for j > 0, if the two eges from v j in R2 point to w 1, w 2, then φ(v j ) is any vertex of G whose two outgoing eges point to φ(w 1 ), φ(w 2 ). It is straightforwar to verify that φ is an isomorphic embeing of R2 in G. Remark 4.1. The theory T has many countable moels which are not isomorphic to R2. To cite a specific example, let G be the graph whose vertex set V (G) is the set of all countable or finite roote binary trees whose eges are labele by natural numbers, such that all but finitely many eges are labele with the successor of their parent s label. For any such tree T with more than one noe, the two outgoing eges from T in G point to its left an right subtrees. 5 Subgraph embeings an non-isomorphism theorem for 3 The aim of this section is to characterize, for each finite graph H, the number of embeings of H in a ranom sample from PA. The non-isomorphism theorem for > 2, Theorem 1.3, will be erive as an easy corollary. Definition 5.1. The orere arboricity of an unirecte graph G is the minimum k such that G amits a vertex orering with the following property: for each vertex v V (G), there are at most k eges connecting v to its preecessors. We will enote the orere arboricity of G by η(g). If G amits such a vertex orering, an if we arbitrarily color the eges from each vertex v to its preecessors with istinct colors from a set of η(g) colors, then the color classes constitute a partition of G s ege set into η(g) acyclic subgraphs, so the orere

8 arboricity of G is boune below by the arboricity. The orere arboricity can be strictly greater than the arboricity, e.g. the ege set of a 4-clique K 4 may be partitione into two isjoint paths, but η(k 4 ) = 3 since the last vertex in any orering is joine to its preceessors by three eges. Theorem 5.1. For a finite graph H, let K H enote the union of all subgraphs of H which have minimum egree 3. If G is a ranom sample from PA, then: 1. If η(h) >, there are no embeings of H in G. 2. If η(h) an K = H then, with probability 1, the number of embeings of H in G is finite. In fact, the expecte number of embeings of H in G is finite an positive. 3. If η(h) an K H then, with probability 1, the number of embeings of H in G is either zero or infinite. Proof. If H is any finite subgraph of G an we orer the vertices of H accoring to their arrival orer, then each vertex has at most eges to its preecessors, which proves that η(h) for any finite subgraph of G. Conversely, if H is a finite graph with η(h), let us label the vertices of H with the numbers 1, 2,..., V (H) in such a way that each ege is joine to its preecessors by at most eges, It is easy to see from the efinition of PA that there is a positive probability the inuce subgraph of G on vertex set {1, 2,..., V (H) } is precisely H (since any eges from {1, 2,..., V (H) } that on t contribute to the embeing of H can attach to vertex 0 of G). If η(h) an K = H, we have alreay shown that the expecte number of embeings of H in G is positive. The fact that it is finite is containe in the following lemma. Lemma 5.1. If K is a graph of minimum egree 3, then the expecte number of embeings of K in G is finite. This lemma forms the crux of the theorem; the complete proof is given in the full version of this paper. The basis of the proof is the observation that, while any two vertices in G almost surely have infinitely many common neighbors, any three vertices in G almost surely have finitely many common neighbors. Informally, this is because any three vertices have egrees θ( t) when vertex t is ae, an so it links to all three with probability θ(t 3/2 ); summing over all t then gives a finite expecte value. Making this precise, however, requires ealing with the conitioning on the egrees, which also pose ifficulties in the proof of Theorem 4.2. To exten this argument to embeings of a graph K of minimum egree 3, we ismantle K by removing one noe at a time, controlling the number of embeings in this ismantling through a boun compose of monomials over the ranom variables X t efine in Section 3. Bouning the expectations of such monomials requires a elicate argument by inuction over the set of all monomials, orere by a ominance orering. (For the etails of the proof, incluing the efinition of the monomial orering, we refer the reaer to the full version of the paper.) We now complete the final case in the proof of Theorem 5.1, when η(h) an K H. We claim that if G contains an embee copy of K, then the number of embeings of H in G is infinite with probability 1. The proof is by inuction on the number of vertices in H \ K. By assumption, K H so there exists a vertex v H whose egree is less than 3. By the inuction hypothesis or by the assumption that K embes in G, we may assume that H \ {v} embes in G. Now by Proposition 4.2, the number of embeings of H in G is infinite with probability 1 (since this proposition asserts that the event there are only a finite number of ways to exten the embeing of H \{v} has unconitional probability 0, an here we re conitioning on a positive-probability event). Note that this establishes case 3, an conclues the proof of the Theorem. A non-isomorphism theorem for > 2. The non-isomorphism theorem for 3 (Theorem 1.3) follows easily from Theorem 5.1. Let N 0 > 0 be the expecte number of istinct embeings of K 4 in G, choose N > N 0. an let K enote the graph consisting of N isjoint copies of K 4. We now consier the probability that G contains a copy of K as a subgraph. Theorem 5.1 asserts that the expecte number of copies of K is positive, an hence there is a positive probability that G contains a copy of K. On the other han, since N > N 0, Markov s inequality ensures that the probability of fining N istinct embeings of K 4 is less than 1, an this implies that the probability that G contains a copy of K is less than 1. Thus the property G contains K as a subgraph is an isomorphism-invariant property, whose truth value has a positive probability of istinguishing two inepenent ranom samples from PA. Proof of Theorem 1.4. The graph property specifie in the previous paragraph is expressible by a firstorer formula φ(g). We claim now that 0 < lim n Pr G PA (n) (φ(g)) < 1. The limit is greater than zero for the same reason as before: Theorem 5.1 implies that G contains K as a

9 subgraph with positive probability. It is also easy to see that Pr G PA (φ(g)) Pr (n) G PA (φ(g)), since a ranom sample from PA is the union of a chain of graphs whose n-th member is a ranom sample from PA (n), an φ is a monotone property. Now the fact that φ(g) is boune away from 1, for graphs of finite size, follows from the fact that Pr G PA (φ(g)) < 1. References [1] W. Aiello, F. Chung, L. Lu. Ranom evolution of massive graphs. Hanbook of Massive Data Sets, (Es. James Abello et al.), Kluwer, 2002, pages [2] Reka Albert an Albert-Laszlo Barabasi. Statistical mechanics of complex networks, Reviews of Moern Physics 74, 47 (2002). [3] A.-L. Barabasi an R. Albert, Emergence of scaling in ranom networks. Science, 286:509, October [4] Aristotle, Physics 239b11-13 [5] P. Billingsley. Probability an Measure. John Wiley an Sons, New York, 3r eition, (1995). [6] B. Bollobas, C. Borgs, J. Chayes, an O. Rioran. Directe scale-free graphs. Proceeings of the 14th ACM- SIAM Symposium on Discrete Algorithms (2003), [7] B. Bollobas. Mathematical results on scalefree ranom graphs. pre-print at [8] B. Bollobas an O. Rioran. The iameter of scale-free graphs. Combinatorica, To appear. [9] B. Bollobas an O. Rioran. Robustness an Vulnerability of Scale-Free Ranom Graphs. Internet Mathematics 1:1(2003) [10] B. Bollobas, Oliver Rioran, Joel Spencer, G. E. Tusnay. The egree sequence of a scale-free ranom graph process. Ranom Structure an Algorithms 18(2001) [11] A. Bonato an J. Janssen. Infinite limits of copying moels of the web graph. Internet Mathematics, 1:2(2003) [12] A. Bonato an J. Janssen. Limits an power laws of moels for the web graph an other networke information spaces, Proceeings of Combinatorial an Algorithmic Aspects of Networking, [13] P. Cameron. The ranom graph. In Algorithms an Combinatorics 14 (R.L. Graham an J. Nesetril, es.), Springer Verlag, New York (1997) [14] C. Cooper, A. M. Frieze. A general moel of web graphs. Proceeings of 9th European Symposium on Algorithms, 2001, [15] F. Chung, L. Lu an V. Vu. Eigenvalues of Ranom Power law Graphs Annals of Combinatorics 7(2003) [16] P. Eros an A. Renyi. On the Evolution of Ranom Graphs. Mat. Kutato Int. Kozl 5 (1960), [17] Paul Erös an Alré Rényi. Asymetric graphs. Acta Math. Aca. Sci. Hung., 14: , [18] Michalis Faloutsos, Petros Faloutsos, Christos Faloutsos. On Power-law Relationships of the Internet Topology. Proc. SIGCOMM 1999, [19] Gilbert, E.N., Ranom graphs, Annals of Mathematical Statistics 30(1959), [20] J. Kleinberg, S.R. Kumar, P. Raghavan, S. Rajagopalan, A. Tomkins. The Web as a graph: Measurements, moels an methos. Proc. International Conference on Combinatorics an Computing, [21] Ravi Kumar, Prabhakar Raghavan, Srihar Rajagopalan, an Anrew Tomkins. Trawling the web for emerging cyber-communities. In Proceeings of the Eighth International Worl Wie Web Conference, [22] Ravi Kumar, Prabhakar Raghavan, Srihar Rajagopalan, Anrew Tomkins: Extracting Large-Scale Knowlege Bases from the Web. VLDB 1999: [23] R. Kumar, P. Raghavan, S. Rajagopalan, D. Sivakumar, A. Tomkins, an E. Upfal. Stochastic moels for the Web graph. Proc. 41st IEEE Symp. on Founations of Computer Science, 2000, pp [24] M. Mihail an C. H. Papaimitriou. On the eigenvalue power law. In Proceeings of RANDOM [25] M. Mihail, C. Papaimitriou, an A. Saberi. On Certain Connectivity Properties of the Internet Topology. Proc. 44th IEEE FOCS [26] M. Newman. The structure an function of networks. Comput. Phys. Commun., 147:40 45, [27] P. Raghavan an R. Motwani, Ranomize Algorithms, Cambrige University Press, [28] J. Spencer an S. Shelah. The strange logic of ranom graphs. Spinger A Bouning E[ X n X nj ] To estimate E( X n X nj ), we transform it into a telescoping sum: E( X n X nj ) = E(X 2 n j 2X nj X n + X 2 n X nj ) = X 2 n j 2X nj E(X n X nj ) + E(X 2 n X nj ) = X 2 n j 2X 2 n j + E(X 2 n X nj ) = E(X 2 n X nj ) X 2 n j = n 1 X k=n j E(X 2 k+1 X nj ) E(X 2 k X nj ). We boun the sum on the right sie term-by-term, using the following computation. Let Z k = k+1 (i) k (i); this is a sum of inepenent Bernoulli ranom variables, each with mean k (i)/2k. Writing k for k (i), a simple computation yiels hence E(Z k ) = k /2k E(Z 2 k) = k /2k + ( 1) 2 k/4 2 k 2 E( 2 k+1 k ) = E(Z 2 k k ) + 2 k E(Z k k ) + 2 k

10 E(X 2 k+1 X k ) < 1 = k /2k + = k /2k k + 1 < k /2k k < = E(X 2 k+1 X nj ) = This means that = 1 c k+1 ck 2kc 2 k+1 ck 2kc 2 k+1 ck 2kc 2 k+1 ck 2kc k+1 E( X n X nj ) < < ««2 k k k/2k k 4k 1 2 4k 2 «2 2 k «2 E( 2 k+1 X k ) «X k k «2 k «2 ck c k+1 «X k + Xk 2 «E(X k X nj ) + E(Xk X 2 nj ) «X nj + E(X 2 k X nj ) 0 n 1 c k 2kc k+1 k=n j X c k 2kc k+1 k=n j < (C/ n j)x nj. 1 A X nj 1 A X nj «2 X 2 k for some constant C, using the fact that each term of the infinite sum is O(k 3/2 ).