Preferential attachment combined with random number of choices

Abstract

Abstract. We study the asymptotic behavior of the maximal degree in the degree distribution in an evolving tree model combining local choice and Mori's preferential attachment. In the considered model, the random graph is constructed in the following way. At each step, a new vertex is introduced. Then, we connect it with one of d possible neighbors, which are sampled from the set of the existing vertices with probability proportional to their degrees plus some parameter β > −1. The number d will be randomly selected and the vertex with the largest degree is chosen. It is known that the maximum of the degree distribution for non-random d > 2 has linear behavior and, for d = 2, asymptotically equals to n/ ln n up to a constant factor. We prove that if Ed < 2 + β, the maximal degree has sublinear behavior with the power Ed/(2 + β) (as in the preferential attachment without choice), if Ed > 2 + β, it has linear behavior and if Ed = 2 + β the maximal degree is of order n/ ln n. The proof combines standard preferential attachment approaches with martingales and stochastic approximation techniques. We also use stochastic approximation results to get a multidimensional central limit theorem for numbers of vertices with fixed degrees.

Preferential attachment with fitness dependent choice

“…The result is similar to the Theorem 1.1 of [Mal18] and shows that the addition of fitness to the choice from the sample does not affect the type of asymptotic of the maximal degree. Let us provide an outline of the proof.…”

“…In this modification, we consider the sample of d independently chosen vertices and then choose one of them by some rule. Two types of rules have been considered: degree base rule (see, e.g., [HJ16,Mal18]) and location (or fitness) based choice (see, e.g., [GLY19,HJY20]). In the present work, we consider choice based on both degree and fitness.…”

“…The other argument we would use is the persistent hub type of argument (see, e.g., [Gal16] and Proposition 1.2 in [Mal18]). This argument is based on the following urn model property.…”