Asymptotic behavior and distributional limits of preferential attachment graphs

Abstract

We give an explicit construction of the weak local limit of a class of preferential attachment graphs. This limit contains all local information and allows several computations that are otherwise hard, for example, joint degree distributions and, more generally, the limiting distribution of subgraphs in balls of any given radius k around a random vertex in the preferential attachment graph. We also establish the finite-volume corrections which give the approach to the limit. . This reprint differs from the original in pagination and typographic detail. 1 2 BERGER, BORGS, CHAYES AND SABERI the paper [11] also introduces a notion of graph limits for sparse graphs with bounded degrees in terms of graph homomorphisms; using expansion methods from mathematical physics, Borgs et al.[10] establishes some general results on this type of limit for sparse graphs. Another recent work [8] concerns limits for graphs which are neither dense nor sparse in the above senses; they have average degrees which tend to infinity. Earlier, a notion of a weak local limit of a sequence of graphs with bounded degrees was given by Benjamini and Schramm [5] (this notion was in fact already implicit in [3]). Interestingly, it is not hard to show that the Benjamini-Schramm limit coincides with the limit defined via graph homomorphisms in the case of sparse graphs of bounded degree; see [16] for yet another equivalent notion of convergent sequences of graphs with bounded degrees.As observed by Lyons [21], the notion of graph convergence introduced by Benjamini and Schramm is meaningful even when the degrees are unbounded, provided the average degree stays bounded. Since the average degree of the Barabási-Albert graph is bounded by construction, it is therefore natural to ask whether this graph sequence converges in the sense of Benjamini and Schramm.In this paper, we establish the existence of the Benjamini-Schramm limit for the Barabási-Albert graph by giving an explicit construction of the limit process, and use it to derive various properties of the limit. Our results cover the case of both uniform and preferential attachments graphs. 1 Moreover, our methods establish the finite-volume corrections which give the approach to the limit.Our proof uses a representation, which we first introduced in [6], to analyze processes that model the spread of viral infections on preferential attachment graphs. Our representation expresses the preferential attachment model process as a combination of several Pólya urn processes. The classic Pólya urn model was of course proposed and analyzed in the beautiful work of Pólya and Eggenberger in the early twentieth century [15]; see [14] for a basic reference. Despite the fact that our Pólya urn representation is a priori only valid for a limited class of preferential attachment graphs, we give an approximating coupling which proves that the limit constructed here is the limit of a much wider class of preferential attachment graphs.Our alternative representation contains much more independence than previous repr...

On the Influence of the Seed Graph in the Preferential Attachment Model

“…For two graphs G and H we write G = H if G and H are isomorphic, and we use the same notation for rooted graphs. Recalling the definition of the Benjamini-Schramm limit (see [Definition 2.1., Berger et al [2014]]), we want to prove that…”

“…The notion of graph limits is more powerful than the one considered in the previous paragraph as it can, for example, distinguish between models having different limiting degree distributions. The weak local limit of the preferential attachment graph was first studied in the case of trees in Rudas et al [2007] using branching process techniques, and then later in general in Berger et al [2014] using Pólya urn representations. These papers show that PA(n, S 2 ) tends to the so-called Pólya-point graph in the weak local limit sense, and our first theorem utilizes this result to obtain the same for an arbitrary seed:…”

“…Theorem 1 For any tree T the weak local limit of PA(n, T ) is the Pólya-point graph described in Berger et al [2014] with m = 1.…”

Exponential Bounds on Graph Enumerations from Vertex Incremental Characterizations

“…These trees are naturally less accurate in terms of giving a precise upper bound, but are much more straightforward to enumerate. This constructive focus is particularly relevant in light of recent results on the preferential attachment model [2,3].…”

“…Nakano et al [23] used the vertex incremental characterization of distance-hereditary graphs to construct corresponding DH-trees. In this paper, we generalize their characterization and introduce the notion of vertex incremental trees, which encode the vertex incremental operations used to construct a given graph and can be applied to other graph classes 3 . DEFINITION 2.3.…”