Robustness analysis of bimodal networks in the whole range of degree correlation

TL;DR

This paper provides an exact analytical study of how degree correlation affects the robustness of bimodal networks under different node removal strategies, revealing that negative correlation enhances network resilience.

Contribution

It introduces a comprehensive analysis of bimodal networks with degree correlation, quantifying how Pearson coefficient influences percolation thresholds and robustness against failures.

Findings

Percolation threshold decreases monotonically with Pearson coefficient.

Negative degree correlation networks are more robust against random failures.

Giant component size declines rapidly in positively correlated networks under failure.

Abstract

We present exact analysis of the physical properties of bimodal networks specified by the two peak degree distribution fully incorporating the degree-degree correlation between node connection. The structure of the correlated bimodal network is uniquely determined by the Pearson coefficient of the degree correlation, keeping its degree distribution fixed. The percolation threshold and the giant component fraction of the correlated bimodal network are analytically calculated in the whole range of the Pearson coefficient from $-1$ to $1$ against two major types of node removal, which are the random failure and the degree-based targeted attack. The Pearson coefficient for next-nearest-neighbor pairs is also calculated, which always takes a positive value even when the correlation between nearest-neighbor pairs is negative. From the results, it is confirmed that the percolation threshold is a monotonically decreasing function of the Pearson coefficient for the degrees of nearest-neighbor pairs increasing from $-1$ and $1$ regardless of the types of node removal. In contrast, the node fraction of the giant component for bimodal networks with positive degree correlation rapidly decreases in the early stage of random failure, while that for bimodal networks with negative degree correlation remains relatively large until the removed node fraction reaches the threshold. In this sense, bimodal networks with negative degree correlation are more robust against random failure than those with positive degree correlation.