Predicting percolation thresholds in networks

TL;DR

This paper evaluates various analytical methods for estimating percolation thresholds in networks without simulations, highlighting the effectiveness of eigenvalue-based bounds and their limitations across different network types.

Contribution

It systematically compares multiple prediction indicators on synthetic and real networks, revealing their strengths and weaknesses in estimating percolation thresholds.

Findings

Inverse of the largest eigenvalue of the non-backtracking matrix often provides a tight lower bound.

In over 40% of cases, naive degree distribution moments outperform eigenvalue-based indicators.

All indicators perform poorly for networks with high percolation thresholds, indicating limited robustness prediction.

Abstract

We consider different methods, that do not rely on numerical simulations of the percolation process, to approximate percolation thresholds in networks. We perform a systematic analysis on synthetic graphs and a collection of 109 real networks to quantify their effectiveness and reliability as prediction tools. Our study reveals that the inverse of the largest eigenvalue of the non-backtracking matrix of the graph often provides a tight lower bound for true percolation threshold. However, in more than 40% of the cases, this indicator is less predictive than the naive expectation value based solely on the moments of the degree distribution. We find that the performance of all indicators becomes worse as the value of the true percolation threshold grows. Thus, none of them represents a good proxy for robustness of extremely fragile networks.