Critical Phase in Complex Networks: a Numerical Study

TL;DR

This paper investigates phase transitions in bond percolation across various complex structures, introducing a fractal exponent to characterize the critical phase, and demonstrates its presence in several non-traditional network models.

Contribution

It introduces a fractal exponent for analyzing critical phases and applies it to diverse nonamenable graphs and growing networks, revealing new insights into percolation phenomena.

Findings

Critical phase exists in nonamenable graphs and certain growing networks.

The fractal exponent effectively characterizes the critical phase.

Percolation behavior varies significantly between different network types.

Abstract

We compare phase transition and critical phenomena of bond percolation on Euclidean lattices, nonamenable graphs, and complex networks. On a Euclidean lattice, percolation shows a phase transition between the nonpercolating phase and percolating phase at the critical point. The critical point is stretched to a finite region, called the critical phase, on nonamenable graphs. To investigate the critical phase, we introduce a fractal exponent, which characterizes a subextensive order of the system. We perform the Monte Carlo simulations for percolation on two nonamenable graphs - the binary tree and the enhanced binary tree. The former shows the nonpercolating phase and the critical phase, whereas the latter shows all three phases. We also examine the possibility of critical phase in complex networks. Our conjecture is that networks with a growth mechanism have only the critical phase and the percolating phase. We study percolation on a stochastically growing network with and without a preferential attachment mechanism, and a deterministically growing network, called the decorated flower, to show that the critical phase appears in those models. We provide a finite-size scaling by using the fractal exponent, which would be a powerful method for numerical analysis of the phase transition involving the critical phase.