Universality for the Distance in Finite Variance Random Graphs

Abstract

We generalize the asymptotic behavior of the graph distance between two uniformly chosen nodes in the configuration model to a wide class of random graphs. Among others, this class contains the Poissonian random graph, the expected degree random graph and the generalized random graph (including the classical Erdős-Rényi graph).In the paper we assign to each node a deterministic capacity and the probability that there exists an edge between a pair of nodes is equal to a function of the product of the capacities of the pair divided by the total capacity of all the nodes. We consider capacities which are such that the degrees of a node have uniformly bounded moments of order strictly larger than two, so that, in particular, the degrees have finite variance. We prove that the graph distance grows like log ν N , where the ν depends on the capacities and N denotes the size of the graph. In addition, the random fluctuations around this asymptotic mean log ν N are shown to be tight. We also consider the case where the capacities are independent copies of a positive random with P ( > x) ≤ cx 1−τ , for some constant c and τ > 3, again resulting in graphs where the degrees have finite variance.The method of proof of these results is to couple each member of the class to the Poissonian random graph, for which we then give the complete proof by adapting the arguments of van der Hofstad et al. (Random Struct. Algorithms 27(2):76-123, 2005).

A Phase Transition for the Diameter of the Configuration Model

“…Similar results for models with conditionally independent edges exist, see e.g. [4,9,14,24]. Thus, for these classes of models, distances are quite well understood.…”

“…We note (see e.g., [19, (4.34)]) that for any two sets of vertices A, B, we have that 14) where, for any A ⊆ {1, . .…”

Asymptotic equivalence and contiguity of some random graphs

“…Example 3.6. van den Esker, van der Hofstad and Hooghiemstra [9] study a minor variation of the construction in Example 3.5; they let Λ 1 , . .…”

“…Assume that P(Λ 1 > t) = o(t −2 ) (which is the case in [9]). Then, just as (3.11) follows from (3.13),…”

Asymptotic analysis of network structures: degree-degree correlations and directed paths

“…This coupling is based on the Kantorovich-Rubinstein distance between two probability measures (see, e.g., [91]), and has the advantage of being uniformly accurate for a considerably longer time than existing constructions. Specifically, the coupling holds for a number of steps in the graph exploration process equivalent to discovering n 1− nodes, for arbitrarily small > 0, compared to a constant number of nodes in [68], n 1/2− nodes in [65] and [40, Theorem 2.2.2], or n 1/2+δ nodes, for a very small δ > 0, in [77,80]. Moreover, the coupled branching process has a deterministic offspring distribution that does not depend on n or the degree sequences, avoiding the need to consider intermediate tree constructions.…”