On the evolution of scale-free graphs

TL;DR

This paper analyzes how scale-free networks evolve as edges are added, revealing different spanning cluster formation behaviors depending on the degree exponent gamma, using a branching process approach.

Contribution

It introduces a theoretical framework for understanding the evolution of scale-free graphs and characterizes the phase transition behavior based on gamma values.

Findings

Spanning cluster forms abruptly for gamma > 3.

For 2 < gamma < 3, spanning cluster formation is gradual with double peaks.

Scaling forms for cluster size distributions are derived.

Abstract

We study the evolution of random graphs where edges are added one by one between pairs of weighted vertices so that resulting graphs are scale-free with the degree exponent $\gamma$. We use the branching process approach to obtain scaling forms for the cluster size distribution and the largest cluster size as functions of the number of edges $L$ and vertices $N$. We find that the process of forming a spanning cluster is qualitatively different between the cases of $\gamma>3$ and $2<\gamma<3$. While for the former, a spanning cluster forms abruptly at a critical number of edges $L_c$, generating a single peak in the mean cluster size $<s>$ as a function of $L$, for the latter, however, the formation of a spanning cluster occurs in a broad range of $L$, generating double peaks in $<s>$.