Distances and large deviations in the spatial preferential attachment model

Abstract

We investigate two asymptotic properties of a spatial preferential-attachment model introduced by E. Jacob and P. Mörters [29]. First, in a regime of strong linear reinforcement, we show that typical distances are at most of doubly-logarithmic order. Second, we derive a large deviation principle for the empirical neighbourhood structure and express the rate function as solution to an entropy minimisation problem in the space of stationary marked point processes.An elegant approach to incorporate clustering into PA models is to use geometry. We embed the network nodes into Euclidean or hyperbolic space and make the PA mechanism aware of spatial distances. Thus, nearby nodes are likely to be connected, thereby inducing clustering [31]. In this sense, spatial preferential attachment models (S-PAMs) combine the virtues of PA models and classical geometric random graphs [38], which exhibit strong local clustering but are not scale-free.The literature offers a variety of definitions and results for spatial PA models [1,11,12,23,24,[29][30][31][32][33][34][35]. In the present paper, we investigate typical distances and large deviation principles (LDPs) in the model introduced in [29], as illustrated in Figure 1. Our methods are fairly robust and we believe that in essence our analysis could be transferred to many of the other architectures mentioned above.Firstly, we verify an intriguing conjecture in [32, Remark 3] about typical distances in the largest connected component of the S-PAM. As made precise in Theorem 2.1 below, the asymptotic behaviour depends on both the strength of the preferential attachment as well as on the influence of vertex distances on the connection probability. This is in contrast to the situation for the asymptotic degree distribution. Indeed, here only the strength of the preferential attachment but not the geometry affects the power-law exponent [31, Remark 1].There are a number of small-world results for complex networks endowed with a geometric structure already available [13,24,28,36,39]. Our work complements these earlier investigations naturally by providing a distance result in a spatial model with preferential attachment in the ultra-small regime of doubly-logarithmic typical distances. Our proof shows that, in this regime, the geometry of the underlying space has an influence on the length of shortest paths in the graph. However, the geometry does not change their fundamental architecture as it can already be observed in non-geometric PA models [7,15,19]. Secondly, we establish an LDP for the in-degree evolution and the evolution of the neighbourhood structure of a typical vertex. The most fundamental building blocks for this result are the process-level LDP of a marked Poisson point process [26] and the contraction principle. The rate function is expressed via a constraint minimisation problem of the specific relative entropy. Loosely speaking, large deviations from the neighbourhood structure are induced by stationary modifications of the original Poisson point process of vertices. ...

Chemical distance in geometric random graphs with long edges and scale-free degree distribution

“…Our results show that, as in the soft Boolean model, we have in the age-dependent random connection model that ultrasmallness fails if γ < δ δ+1 . If γ > δ δ+1 we get a lower bound matching that of [22] and we get the precise asymptotics for the chemical distance as stated in (2).…”

“…In fact, the following lemma shows that for a powerful vertex with mark t and a suitable vertex with a sufficiently smaller mark, the probability that there exist no connector which neighbours each of the two vertices is decaying exponentially fast as the mark t gets small. This is a corollary of [22] and follows with the same calculations as in [18,Lemma 3.1]. We now fix for the rest of the section…”

“…As (t/s) γ is the asymptotic order of the expected degree at time t of a vertex born at time s ↓ 0 this infinite graph model mimics the behaviour of spatial preferential attachment networks [2,23]. An upper bound for the chemical distance for spatial preferential attachment is given by Hirsch and Mönch in [22], but lower bounds are not known. Our results show that, as in the soft Boolean model, we have in the age-dependent random connection model that ultrasmallness fails if γ < δ δ+1 .…”

“…Suppose Assumption 1.2 holds for γ > δ δ+1 . Then for any vertex x there exists a path with no more than To prove this result, we rely on a strategy introduced in [22]. Since the vertices of G are given by the points of a Poisson process, the most powerful vertex inside a box with volume of order |x| d around the midpoint between 0 and x typically has a mark smaller than |x| −d log |x|.…”

“…While the layer L 1 only contains vertices with very small mark, vertices with larger and larger marks are included in layers with larger index. More precisely, as in [22] we set where η = (γ − (α 2 − α 1 γ)δ) ∧ (α 2 − α 1 ) > 0. As the vertex set X is a Poisson process, by Lemma 3.1 for a given vertex in layer L k+1 there exists with high probability a suitable vertex in layer L k such that both vertices are connected via a connector with high probability.…”