Random walk attachment graphs

TL;DR

This paper investigates the properties of random walk attachment graphs, showing how their structure can differ significantly from preferential attachment graphs, especially in cases with short random walks.

Contribution

It demonstrates that fixed-length random walk attachment graphs can exhibit different degree distributions compared to traditional preferential attachment models.

Findings

Proportion of degree-1 vertices tends to 1 when random walk length is 1.

Graphs with fixed random walk length can differ markedly from preferential attachment graphs.

Abstract

We consider the random walk attachment graph introduced by Saram\"{a}ki and Kaski and proposed as a mechanism to explain how behaviour similar to preferential attachment may appear requiring only local knowledge. We show that if the length of the random walk is fixed then the resulting graphs can have properties significantly different from those of preferential attachment graphs, and in particular that in the case where the random walks are of length 1 and each new vertex attaches to a single existing vertex the proportion of vertices which have degree 1 tends to 1, in contrast to preferential attachment models. AMS 2010 Subject Classification: Primary 05C82. Key words and phrases:random graphs; preferential attachment; random walk.