Random networks with sublinear preferential attachment: Degree evolutions

TL;DR

This paper introduces a dynamic model of random networks with sublinear preferential attachment, analyzing degree distributions and phase transitions in the evolution of maximum degree vertices.

Contribution

It provides a strong limit law for degree distribution and characterizes phase transitions in the emergence of dominant vertices based on attachment preferences.

Findings

In strong preference cases, a single vertex becomes dominant.

In weak preference cases, the maximum degree vertex changes infinitely often.

A phase transition occurs at the square root attachment probability.

Abstract

We define a dynamic model of random networks, where new vertices are connected to old ones with a probability proportional to a sublinear function of their degree. We first give a strong limit law for the empirical degree distribution, and then have a closer look at the temporal evolution of the degrees of individual vertices, which we describe in terms of large and moderate deviation principles. Using these results, we expose an interesting phase transition: in cases of strong preference of large degrees, eventually a single vertex emerges forever as vertex of maximal degree, whereas in cases of weak preference, the vertex of maximal degree is changing infinitely often. Loosely speaking, the transition between the two phases occurs in the case when a new edge is attached to an existing vertex with a probability proportional to the root of its current degree.