The vertex attack tolerance of complex networks.

The purpose of this work is four-fold: (1) We propose a new measure of network resilience in the face of targeted node attacks, vertex attack tolerance, represented mathematically as tau(G) = min(S subset of V) |S|/|V-S-C-max(V-S)|+1, and prove that for d-regular graphs tau(G) = Theta(Phi(G))where Phi(G) denotes conductance, yielding spectral bounds as corollaries. (2) We systematically compare tau(G) to known resilience notions, including integrity, tenacity, and toughness, and evidence the dominant applicability of tau for arbitrary degree graphs. (3) We explore the computability of tau, first by establishing the hardness of approximating unsmoothened vertex attack tolerance (tau) over cap (G) = min(S subset of V) |S|/|V-S-C-max(V-S)| under various plausible computational complexity assumptions, and then by presenting empirical results on the performance of a betweenness centrality based heuristic algorithm applied not only to tau but several other hard resilience measures as well. (4) Applying our algorithm, we find that the random scale-free network model is more resilient than the Barabasi-Albert preferential attachment model, with respect to all resilience measures considered.