Robustness of Complex Networks with Implications for Consensus and Contagion

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.

💡 Research Summary

The paper investigates a graph‑theoretic property called robustness, which has become central to the analysis of several network dynamics such as resilient consensus, contagion processes, and bootstrap percolation. Robustness, formalized as r‑robustness, requires that for every non‑empty vertex subset S either (i) the external neighborhood N(S) contains at least r vertices, or (ii) the size of S itself is at least r. This condition is strictly stronger than ordinary vertex‑connectivity (κ) and minimum degree (δ); one can construct graphs with high κ and δ that nevertheless fail to be r‑robust.

The authors focus on three canonical random‑graph models that capture many real‑world complex networks:

1. Erdős‑Rényi G(n,p) – The paper derives the exact threshold for r‑robustness as
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