Critical and Nonpercolating Phases in Bond Percolation on the Song-Havlin-Makse Network

TL;DR

This paper analyzes bond percolation on the Song-Havlin-Makse network, revealing distinct phases depending on the network's fractal or small-world nature, with analytical and simulation results aligning closely.

Contribution

It provides the first analytical characterization of percolation phases on the SHM network, highlighting the impact of network dimensionality on percolation behavior.

Findings

Fractal SHM networks remain nonpercolating for all p<1.

Small-world SHM networks exhibit a critical phase with power-law cluster distribution.

Analytical results agree well with Monte Carlo simulations.

Abstract

We investigate bond percolation on the Song-Havlin-Makse (SHM) network, a scale-free tree with a tunable degree exponent and dimensionality. Using a generating function approach, we analytically derive the average size and the fractal exponent of the root cluster for deterministic cases. Our analysis reveals that bond percolation on the SHM network remains in a nonpercolating phase for all $p < 1$ when the network is fractal (i.e., finite-dimensional), whereas it exhibits a critical phase, where the cluster size distribution follows a power-law with a $p$-dependent exponent, throughout the entire range of $p$ when the network is small-world (i.e., infinite-dimensional), regardless of the specific dimensionality or degree exponent. The analytical results are in excellent agreement with Monte Carlo simulations.