The Clustering Coefficient of a Scale-Free Random Graph

TL;DR

This paper analyzes the clustering coefficient in a scale-free random graph model, showing it decreases proportionally to log n/n, with differences depending on whether a constant is added to degree-based probabilities.

Contribution

It provides an asymptotic analysis of the clustering coefficient in a generalized scale-free graph model, extending previous results to include a positive constant.

Findings

Clustering coefficient asymptotically proportional to log n/n with positive constant.

When the constant is zero, clustering coefficient scales as (log n)^2/n.

Results extend understanding of clustering in scale-free networks.

Abstract

We consider a random graph process in which, at each time step, a new vertex is added with m out-neighbours, chosen with probabilities proportional to their degree plus a strictly positive constant. We show that the expectation of the clustering coefficient of the graph process is asymptotically proportional to log n/n. Bollob\'as and Riordan have previously shown that when the constant is zero, the same expectation is asymptotically proportional to ((log n)^2)/n.