On Bollobás‐Riordan random pairing model of preferential attachment graph

Abstract

Bollobás-Riordan random pairing model of a preferential attachment graph G n m is studied. Let {W j } j ≤ mn + 1 be the process of sums of independent exponentials with mean 1. We prove that the degrees of the first n m ≔ n m m+2 − vertices are jointly, and uniformly, asymptotic to, and that with high probability (whp) the smallest of these degrees is n (m+2) 2m , at least. Next we bound the probability that there exists a pair of large vertex sets without connecting edges, and apply the bound to several special cases. We propose to measure an influence of a vertex v by the size of a maximal recursive tree (max-tree) rooted at v. We show that whp the set of the first n m vertices does not contain a max-tree, and the largest max-tree has size of order n. We prove that, for m > 1, P(G n m is connected) ≥ 1−O((log n) −(m−1)∕3+o( 1) ). We show that the distribution of scaled size of a generic max-tree in G n 1 converges to a mixture of two beta distributions.

Giant descendant trees, matchings, and independent sets in age-biased attachment graphs

“…In this paper we construct a new factorial-type martingale with one argument being the total size of a 'descendant' subtree. This is a generalization of the martingale for δ = 0, found by Pittel [36].…”

“…Fix a positive integer r and let X(t) denote the number of descendants of r at time t. Here r is a descendant of r, and x is a descendant of r if and only if x chooses to attach itself to at least one descendant of r in step x. In other words, if we think of the graph as a directed graph with edges oriented towards the smaller vertices, vertex x is a descendant of r if and only if there is a directed decreasing path from x to r. In [36] X(t) was proposed as an influence measure of vertex r at time t.…”

“…Acan and Hitczenko [1] found an alternative proof, without rate, via a memory game. Pittel [36] used the Bollobás-Riordan pairing model to approximate, with explicit error estimate, the degree sequence of the first n m/(m+2) vertices, m ≥ 1, and proved that, for m > 1, PAM is connected with probability ≈ 1 − O((log n) −(m−1)/3 ). Random walks on PAM have been considered in the work of Cooper and Frieze [14,15].…”

Random plane increasing trees: Asymptotic enumeration of vertices by distance from leaves

“…In a different direction, $$$ {\left\{{T}_n\right\}}_{n\ge 1} $$$ is a well‐known case of the random tree (graph, if cycles allowed) process with “preferential attachment”. See Barabási and Albert [1], Móri [17], [18], Bollobás and Riordan [4], Bollobás [3], van der Hofstad [12], Frieze and Karoński [10], Pittel [20].…”