Asymptotic degree distribution in preferential attachment graph models with multiple type edges

Abstract

We deal with a general preferential attachment graph model with multiple type edges. The types are chosen randomly, in a way that depends on the evolution of the graph. In the N-type case, we define the (generalized) degree of a given vertex as d ¼ ðd 1 ; d 2 ;. .. ; d N Þ; where d k 2 Z þ 0 is the number of type k edges connected to it. We prove the existence of an a.s. asymptotic degree distribution for a general family of preferential attachment random graph models with multi-type edges. More precisely, we show that the proportion of vertices with (generalized) degree d tends to some random variable as the number of steps goes to infinity. We also provide recurrence equations for the asymptotic degree distribution. Finally, we generalize the scale-free property of random graphs to the multi-type case.

Barabási–Albert random graph with multiple type edges and perturbation

“…F n is the identity matrix for every n. It also means that the condition on the irreducibility of F fails in Theorem 1. However, Theorem 2 in [3] describes the asymptotic degree distribution in the nonperturbed version of the model. This is the following: in the multi-type Barabási-Albert random graph for every d ∈ N N we have lim n→∞ X n (d) |V n | = x(d) a.s., where now x(d) is a non-deterministic random variable.…”

“…Now, we can prove our main result on the asymptotic degree distribution of the perturbed Barabási-Albert random graph. We summarize a general graph model with its assumptions (GM1)-(GM5) from [3]. We will see that the perturbed Barabási-Albert model is a special case.…”

“…It is chosen randomly when the edge is born, with a distribution that depends on the current state of the graph and on the types of the edges going out from the neighbours of the new vertex. A general family of preferential attachment random graphs with multi-type edges has been examined in our previous work [3]. Growing networks with two different types of edges can be considered as directed graphs.…”

“…That is, when a new vertex is born, then it is attached to the graph with an edge directed from the new vertex to the already existing ones or directed from the existing vertices to the new one, and this corresponds to the two different types. Directed preferential attachment models were introduced and examined in [5,14], but with different dynamics than in [3]. In this paper we are interested in a version of robustness in preferential attachment graph models with multi-type edges.…”

Asymptotic normality of degree counts in a general preferential attachment model

“…This led to consider some extensions of the classical PA model. For example, in [3] the authors consider a general family of preferential attachment models with multi-type edges and in [1] with edge-steps, while [2,11,12,24] investigate a PA model which mixes PA rules with uniform attachment rules.…”