Relaxation dynamics of maximally clustered networks

TL;DR

This paper investigates how maximally clustered networks relax to unclustered states under different edge dynamics, deriving analytical expressions and identifying a phase transition in network connectivity.

Contribution

It provides the first analytical description of relaxation dynamics and phase transition in maximally clustered networks under two edge rewiring processes.

Findings

Networks undergo a continuous phase transition forming a giant component.

Analytical expressions for degree and clustering evolution are derived.

Phase transition point is calculated using Erdős–Rényi theory.

Abstract

We study the relaxation dynamics of fully clustered networks (maximal number of triangles) to an unclustered state under two different edge dynamics---the double-edge swap, corresponding to degree-preserving randomization of the configuration model, and single edge replacement, corresponding to full randomization of the Erd\H{o}s--R\'enyi random graph. We derive expressions for the time evolution of the degree distribution, edge multiplicity distribution and clustering coefficient. We show that under both dynamics networks undergo a continuous phase transition in which a giant connected component is formed. We calculate the position of the phase transition analytically using the Erd\H{o}s--R\'enyi phenomenology.