Evolution of scale-free random graphs: Potts model formulation

TL;DR

This paper investigates the percolation properties of scale-free random graphs using a Potts model approach, revealing different cluster formation behaviors depending on the degree exponent.

Contribution

It introduces a novel Potts model formulation to analyze bond percolation in scale-free graphs with inhomogeneous interactions, deriving critical exponents and finite-size scaling.

Findings

Giant cluster forms abruptly for λ>3

Gradual giant cluster formation for 2<λ<3

Finite-size effects show double peaks in mean cluster size

Abstract

We study the bond percolation problem in random graphs of $N$ weighted vertices, where each vertex $i$ has a prescribed weight $P_i$ and an edge can connect vertices $i$ and $j$ with rate $P_iP_j$. The problem is solved by the $q\to 1$ limit of the $q$-state Potts model with inhomogeneous interactions for all pairs of spins. We apply this approach to the static model having $P_i\propto i^{-\mu} (0<\mu<1)$ so that the resulting graph is scale-free with the degree exponent $\lambda=1+1/\mu$. The number of loops as well as the giant cluster size and the mean cluster size are obtained in the thermodynamic limit as a function of the edge density, and their associated critical exponents are also obtained. Finite-size scaling behaviors are derived using the largest cluster size in the critical regime, which is calculated from the cluster size distribution, and checked against numerical simulation results. We find that the process of forming the giant cluster is qualitatively different between the cases of $\lambda >3$ and $2 < \lambda <3$. While for the former, the giant cluster forms abruptly at the percolation transition, for the latter, however, the formation of the giant cluster is gradual and the mean cluster size for finite $N$ shows double peaks.