Clarifying How Degree Entropies and Degree-Degree Correlations Relate to Network Robustness

TL;DR

This paper clarifies the relationship between degree entropy, degree correlations, and network robustness, showing that degree entropy alone is insufficient as a robustness proxy, and proposing improved measures for structural robustness analysis.

Contribution

It demonstrates that degree distribution entropy is only a lower bound for robustness and introduces a better measure based on remaining degree entropy and mutual information adjustments.

Findings

Degree distribution entropy is a lower bound for network robustness.

Remaining degree entropy correlates positively with robustness.

Adjusted mutual information better captures structural properties related to robustness.

Abstract

It is often claimed that the entropy of a network's degree distribution is a proxy for its robustness. Here, we clarify the link between degree distribution entropy and giant component robustness to node removal by showing that the former merely sets a lower bound to the latter for randomly configured networks when no other network characteristics are specified. Furthermore, we show that, for networks of fixed expected degree that follow degree distributions of the same form, the degree distribution entropy is not indicative of robustness. By contrast, we show that the remaining degree entropy and robustness have a positive monotonic relationship and give an analytic expression for the remaining degree entropy of the log-normal distribution. We also show that degree-degree correlations are not by themselves indicative of a network's robustness for real networks. We propose an adjustment to how mutual information is measured which better encapsulates structural properties related to robustness.