Generating-function approach for bond percolations in hierarchical networks

TL;DR

This paper investigates bond percolation on hierarchical scale-free networks using a generating function approach, revealing a critical phase with continuously varying exponents and power-law cluster size distributions.

Contribution

It introduces a generating function method to analyze bond percolation in hierarchical networks, showing continuous variation of critical exponents within the critical phase.

Findings

Fractal exponent $oldsymbol{\psi}$ varies continuously with $ ilde{p}$

Cluster size distribution follows a power law $n_s \,\propto\, s^{- au}$

Critical exponent $eta( ilde{p})$ varies from 0.164694 to infinity

Abstract

We study bond percolations on hierarchical scale-free networks with the open bond probability of the shortcuts $\tilde{p}$ and that of the ordinary bonds $p$. The system has a critical phase in which the percolating probability $P$ takes an intermediate value $0<P<1$. Using generating function approach, we calculate the fractal exponent $\psi$ of the root clusters to show that $\psi$ varies continuously with $\tilde{p}$ in the critical phase. We confirm numerically that the distribution $n_s$ of cluster size $s$ in the critical phase obeys a power law $n_s \propto s^{-\tau}$, where $\tau$ satisfies the scaling relation $\tau=1+\psi^{-1}$. In addition the critical exponent $\beta(\tilde{p})$ of the order parameter varies as $\tilde{p}$, from $\beta\simeq 0.164694$ at $\tilde{p}=0$ to infinity at $\tilde{p}=\tilde{p}_c=5/32$.