Robustness of Random Graphs Based on Natural Connectivity

TL;DR

This paper investigates the robustness of random graphs using natural connectivity, showing it increases linearly with average degree and varies with graph size, with implications for network resilience.

Contribution

It provides analytical and numerical analysis of natural connectivity in random graphs, comparing their robustness to regular lattices and deriving critical graph size thresholds.

Findings

Natural connectivity increases linearly with average degree.

Random graphs are more robust than regular lattices below a critical size.

The relationship between robustness and graph type depends on size and degree.

Abstract

Recently, it has been proposed that the natural connectivity can be used to efficiently characterise the robustness of complex networks. Natural connectivity quantifies the redundancy of alternative routes in a network by evaluating the weighted number of closed walks of all lengths and can be regarded as the average eigenvalue obtained from the graph spectrum. In this article, we explore the natural connectivity of random graphs both analytically and numerically and show that it increases linearly with the average degree. By comparing with regular ring lattices and random regular graphs, we show that random graphs are more robust than random regular graphs; however, the relationship between random graphs and regular ring lattices depends on the average degree and graph size. We derive the critical graph size as a function of the average degree, which can be predicted by our analytical results. When the graph size is less than the critical value, random graphs are more robust than regular ring lattices, whereas regular ring lattices are more robust than random graphs when the graph size is greater than the critical value.