A percolation system with extremely long range connections and node dilution

Abstract

We study the very long-range bond-percolation problem on a linear chain with both sites and bonds dilution. Very long range means that the probability $p_{ij}$ for a connection between two occupied sites $i,j$ at a distance $r_{ij}$ decays as a power law, i.e. $p_{ij}=\rho /[r_{ij}^{\alpha }{N}^{1-\alpha }]$ when $0\le \alpha <1$, and $p_{ij}=\rho /[r_{ij}\ln (N)]$ when $\alpha =1$. Site dilution means that the occupancy probability of a site is $0<p_s\le 1$. The behavior of this model results from the competition between long-range connectivity, which enhances the percolation, and site dilution, which weakens percolation. The case $\alpha =0$ with $p_s=1$ is well-known, being the exactly solvable mean-field model. The percolation order parameter $P_{\infty }$ is investigated numerically for different values of $\alpha$, $p_s$ and $\rho$. We show that in the ranges $0\le \alpha \le 1$ and $0<p_s\le 1$ the percolation order parameter $P_{\infty }$ depends only on the average connectivity $\gamma$ of sites, which can be explicitly computed in terms of the three parameters $\alpha$, $p_s$ and $\rho$.

References in corpus (3)

• Structural properties of spatially embedded networks
• Percolation with long-range correlated disorder
• Reaction spreading on percolating clusters

Cited by in corpus (1)

• Bootstrap percolation on spatial networks