Nongrowing Preferential Attachment Random Graphs

Abstract

We consider an edge rewiring process which is widely used to model the dynamics of scale-free weblike networks. This process uses preferential attachment and operates on sparse multigraphs with n vertices and m edges. We prove that its mixing time is optimal and develop a framework which simplifies the calculation of graph properties in the steady state. The applicability of this framework is demonstrated by calculating the degree distribution, the number of self-loops and the threshold for the appearance of the giant component.

Preferential attachment without vertex growth: emergence of the giant component

“…It follows that we can use the random multigraph G α, * n,m to derive the long-run asymptotic properties of the edge-rewiring process with n vertices and m edges. In particular, the result by Pittel [43] (or our Theorem 1.1 and 2.1) confirms Conjecture 4.3 of Hruz and Peter [23] regarding the existence of a giant component.…”

“…To be more precise, this preferential attachment edge-rewiring process starts with a given (arbitrary) initial multigraph G 0 with vertex set [n] and m multiedges, and then proceeds stepwise as follows: a uniform endvertex v of a uniformly chosen edge e is selected, and then e is replaced with the edge {v, w}, where w is chosen with probability proportional to d w +α. This rewiring process (also called simple edge-selection process [23] or edge reconnecting model [45]) converges rapidly to a unique stationary distribution G α,∞ n,m (see [23, Sections 1.1-1.2], [45, Section 2.2], and the 'equilibrium' discussion in [12,11]), which in fact has the same distribution as G α, * n,m (see [45,Lemma 2.1]). It follows that we can use the random multigraph G α, * n,m to derive the long-run asymptotic properties of the edge-rewiring process with n vertices and m edges.…”