Limited path percolation in complex networks

TL;DR

This paper investigates how the stability of network communication is affected by link removal when effective communication requires shortest paths to be within a scaled factor of the original, revealing a new percolation transition.

Contribution

It introduces a novel percolation transition in complex networks considering limited path lengths, supported by analytical and simulation results on different network models.

Findings

A new percolation transition at a specific link removal threshold.

Below the threshold, only a fraction of nodes can communicate.

Above the threshold, most nodes can communicate within limited path lengths.

Abstract

We study the stability of network communication after removal of $q=1-p$ links under the assumption that communication is effective only if the shortest path between nodes $i$ and $j$ after removal is shorter than $a\ell_{ij} (a\geq1)$ where $\ell_{ij}$ is the shortest path before removal. For a large class of networks, we find a new percolation transition at $\tilde{p}_c=(\kappa_o-1)^{(1-a)/a}$, where $\kappa_o\equiv < k^2>/< k>$ and $k$ is the node degree. Below $\tilde{p}_c$, only a fraction $N^{\delta}$ of the network nodes can communicate, where $\delta\equiv a(1-|\log p|/\log{(\kappa_o-1)}) < 1$, while above $\tilde{p}_c$, order $N$ nodes can communicate within the limited path length $a\ell_{ij}$. Our analytical results are supported by simulations on Erd\H{o}s-R\'{e}nyi and scale-free network models. We expect our results to influence the design of networks, routing algorithms, and immunization strategies, where short paths are most relevant.