Scale-free graphs with many edges

TL;DR

This paper analyzes the likelihood of having many edges in a Chung-Lu random graph with heavy-tailed degree distribution, showing that hubs are the main cause of unusually dense graphs.

Contribution

It provides tail estimates for the number of edges in such graphs, highlighting the role of hubs in extreme cases.

Findings

Hubs significantly increase the probability of dense graphs.

Tail estimates quantify the likelihood of large edge counts.

Presence of hubs explains large deviations in edge numbers.

Abstract

We develop tail estimates for the number of edges in a Chung-Lu random graph with regularly varying weight distribution. Our results show that the most likely way to have an unusually large number of edges is through the presence of one or more hubs, i.e.\ vertices with degree of order $n$.