Robustness of Complex Networks with Implications for Consensus and Contagion

TL;DR

This paper investigates the robustness property of complex networks, showing its equivalence to connectivity in common models and highlighting its importance for diffusion processes, while also proving the computational difficulty of assessing robustness.

Contribution

It establishes the equivalence of robustness and connectivity in key random graph models and proves the coNP-completeness of robustness verification.

Findings

Robustness coincides with connectivity in Erdos-Renyi, geometric, and preferential attachment graphs.

Threshold functions for robustness and connectivity are identical in Erdos-Renyi models.

Determining robustness is coNP-complete for any given graph.

Abstract

We study a graph-theoretic property known as robustness, which plays a key role in certain classes of dynamics on networks (such as resilient consensus, contagion and bootstrap percolation). This property is stronger than other graph properties such as connectivity and minimum degree in that one can construct graphs with high connectivity and minimum degree but low robustness. However, we show that the notions of connectivity and robustness coincide on common random graph models for complex networks (Erdos-Renyi, geometric random, and preferential attachment graphs). More specifically, the properties share the same threshold function in the Erdos-Renyi model, and have the same values in one-dimensional geometric graphs and preferential attachment networks. This indicates that a variety of purely local diffusion dynamics will be effective at spreading information in such networks. Although graphs generated according to the above constructions are inherently robust, we also show that it is coNP-complete to determine whether any given graph is robust to a specified extent.