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We study a graph-theoretic property known as robustness, which plays a key role in the behavior of certain classes of dynamics on networks (such as resilient consensus and contagion). This property is much 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. In this paper, we investigate the robustness of common random graph models for complex networks (Erdos-Re&#x0301;nyi, geometric random, and preferential attachment graphs). We show that the notions of connectivity and robustness coincide on these random graph models: the properties share the same threshold function in the Erdos-Re&#x0301;nyi model, cannot be very different in the geometric random graph model, and are equivalent in the preferential attachment model. This indicates that a variety of purely local diffusion dynamics will be effective at spreading information in such networks.

Robustness of complex networks with implications for consensus and contagion