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The Giant in Random Graphs Is Almost Local

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The Giant in Random Graphs Is Almost Local The giant in random graphs is almost local Remco van der Hofstad Department of Mathematics and Computer Science, Eindhoven University of Technology, e-mail: [email protected] Abstract: Local convergence techniques have become a key methodology to study random graphs in sparse settings where the average degree remains bounded. However, many ran- dom graph properties do not directly converge when the random graph converges locally. A notable, and important, random graph property that does not follow from local convergence is the size and uniqueness of the giant component. We provide a simple criterion that guar- antees that local convergence of a random graph implies the convergence of the proportion of vertices in the maximal connected component. We further show that, when this condition holds, the local properties of the giant are also described by the local limit. We apply this novel method to the configuration model as a proof of concept, reproving a result that is well-established. As a side result this proof, we show that the proof also implies the small-world nature of the configuration model. MSC2020 subject classifications: 60K37, 05C81, 82C27, 37A25. Keywords and phrases: Random graphs, Local convergence, Giant component, Configu- ration model. 1. Introduction Local convergence techniques, as introduced by Aldous and Steele [3] and Benjamini and Schramm [13], have become the main methodology to study random graphs in sparse settings where the average degree remains bounded. Local convergence roughly means that the proportion of ver- tices whose neighborhoods have a certain shape converges to some limiting value, which is to be considered as a measure on rooted graphs. We refer to [39, Chapter 2] for background on local convergence. The giant component problem has received enormous attention ever since the very first and seminal results by Erd}osand R´enyi on the Erd}os-R´enyi random graph [33], see also [48, 18] for detailed further results for this model. The simplest form of this question is whether there exists a linear-size connected component or not, in many cases a sharp transition occurs depending on a certain graph parameter. See [6, 32, 50, 38, 39], as well as the references therein, for several books on the topic. This paper combines these two threads by investigating the size of the giant component when the random graph converges locally. Consider a sequence of random graphs (Gn)n≥1, where we will simplify the notation by assuming that Gn = (V (Gn);E(Gn)) is such that jV (Gn)j = n. We let jCmaxj = max jC (v)j (1.1) v2V (Gn) arXiv:2103.11733v1 [math.PR] 22 Mar 2021 denotes the maximal cluster size. When the random graph converges locally to some limiting rooted P measure, then one would expect that also jCmaxj=n −! ζ, where ζ is the survival probability of the local limit. However, while the number of connected components is well behaved in the local topology, the proportion of vertices in the giant is not so nicely behaved. Indeed, since local convergence is all about proportions of vertices whose finite (but arbitrarily large) neighborhoods converge to a limit, there is an enormous gap between surviving locally and being in the giant component. Thus, local convergence is obviously not enough to imply the convergence of the giant. In this paper, we identify what extra condition is needed to imply this natural implication. 2. Asymptotics and properties of the giant In this section, we investigate the behavior of the giant component jCmaxj for random graphs that converge locally. In Section 2.1, we introduce the notion of local convergence in probability. 1 2 In Section 2.2, we study the asymptotic of jCmaxj for random graphs that converge locally in probability, and in Section 2.3 we investigate local properties of the giant. 2.1. Local convergence in probability Local convergence was introduced independently by Aldous and Steele in [3] and Benjamini and Schramm in [13]. The purpose of Aldous and Steele in [3] was to describe the local structure of the so-called `stochastic mean-field model of distance', meaning the complete graph with i.i.d. exponential edge weights. This local description allowed Aldous to prove the celebrated ζ(2) limit of the random assignment problem [1]. Benjamini and Schramm in [13] instead used local weak convergence to show that limits of planar graphs are with probability one recurrent. Since its conception, local convergence has proved a key ingredient in random graph theory. In this section, we provide some basics of local convergence. For more detailed discussions, we refer the reader to [22] or [39, Chapter 2]. Let us start with some definitions. A rooted graph is a pair (G; o), where G = (V (G);E(G)) is a graph with vertex set V (G) and edge set E(G), and o 2 V (G) is a vertex. Further, a rooted or non-rooted graph is called locally finite when each of its vertices has finite degree (though not necessarily uniformly bounded). Two (finite or infinite) graphs G1 = (V (G1);E(G1)) and G2 = (V (G2);E(G2)) are called isomorphic, which we write as G1 ' G2, when there exists a bijection φ: V (G1) 7! V (G2) such that fu; vg 2 E(G1) precisely when fφ(u); φ(v)g 2 E(G2): Similarly, two rooted (finite or infinite) graphs (G1; o1) and (G2; o2) ,wtih Gi = (V (Gi);E(Gi)) for i 2 f1; 2g, are called isomorphic, abbreviated as (G1; o1) ' (G2; o2), when there exists a bijection φ: V (G1) 7! V (G2) such that φ(o1) = o2 and fu; vg 2 E(G1) precisely when fφ(u); φ(v)g 2 E(G2): These notions can be easily adapted to multi-graphs (which we will need to rely on), for which G = (V (G); (xi;j)i;j2V (G)), where xi;j denotes the number of edges between i and j, and xi;i the number of self-loops at i. There, instead, 0 0 the isomorphism φ: V (G) 7! V (G ) is required to satisfy that xi;j = xφ(i),φ(j), where (xi;j)i;j2V (G) 0 0 and (xi;j)i;j2V (G0) are the edge multiplicities of G and G respectively. We let G? be the space of (possibly infinite) connected rooted graphs, where we consider two graphs to be equal when they are isomorphic. Thus, we consider G? as the set of equivalence classes of rooted graphs modulo isomorphisms. The space G? of rooted graphs is a nice metric space, with an explicit metric, see [39, Appendix A] for details. (G) For a rooted graph (G; o), we let Br (o) denote the (rooted) subgraph of (G; o) of all vertices at (G) (G) (G) graph distance at most r away from o. Formally, this means that Br (o) = ((V (Br (o));E(Br (o)); o), where (G) V (Br (o)) = fu: dG(o; u) ≤ rg; (2.1) (G) E(Br (o)) = ffu; vg 2 E(G): dG(o; u); dG(o; v) ≤ rg: Let o 2 V (Gn) be chosen uniformly at random (u.a.r.) in V (Gn). We say that the graph sequence ? (Gn)n≥1 converges locally in probability to a limit (G; o) ∼ µ, when, for every r ≥ 0 and H 2 G?, 1 X 1 P (G) ? fB(Gn)(v)'H?g −! µ(Br (v) ' H ): (2.2) jV (Gn) r v2V (Gn) This means that the subgraph proportions in the random graph Gn are close, in probability, to those given by µ. There are related notions of local convergence, such as local weak convergence, where (2.2) is replaced by the expectation of the left-hand side converging to those on the right hand side, and local almost sure convergence, where the convergence in probability in (2.2) is replaced by convergence almost surely. For our purposes, local convergence in probability is the most convenient, for example since it implies that the neighborhoods of two uniformly chosen vertices are close to being independent (see e.g., [39, Corollary 2.18]), which is central in our proof. 2.2. Asymptotics of the giant Given a random graph sequence Gn that converges in locally probability to (G; o) ∼ µ, one P would expect that jCmaxj=n −! ζ := µ(jC (o)j = 1). However, the proportion of vertices in the 2.2 Asymptotics of the giant 3 largest connected component jCmaxj=n is not continuous in the local convergence topology, as it is a global object. In fact, also jC (on)j=n does not converge in distribution when Gn converges locally in probability to (G; o) ∼ µ. However, local convergence still tells us a useful story about the existence of a giant, as well as its size. Indeed, Corollary 2.1 shows that the upper bound is always valid, while Theorem 2.2 shows that a relative simple condition suffices to yield the lower bound as well: Corollary 2.1 (Upper bound on the giant). Let (Gn)n≥1 be a sequence of graphs whose sizes jV (Gn)j = n tend to infinity. Assume that Gn converges locally in probability to (G; o) ∼ µ. Write ζ = µ(jC (o)j = 1) for the survival probability of the limiting graph (G; o). Then, for every " > 0 fixed, P(jCmaxj ≤ n(ζ + ")) ! 1: (2.3) P In particular, Corollary 2.1 implies that jCmaxj=n −! 0 when ζ = 0, so that there can only be a giant when the local limit has a positive survival probability. Proof. Define X 1 Z≥k = fjC (v)|≥kg: (2.4) v2V (Gn) Assume that Gn converges locally in probability to (G; o) ∼ µ as defined in (2.2).
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