Correlations in complex networks under attack.

For any initially correlated network after any kind of attack where either nodes or edges are removed, we obtain general expressions for the degree-degree probability matrix and degree distribution. We show that the proposed analytical approach predicts the correct topological changes after the attack by comparing the evolution of the assortativity coefficient for different attack strategies and intensities in theory and simulations. We find that it is possible to turn an initially assortative network into a disassortative one, and vice versa, by fine-tuning removal of either nodes or edges. For an initially uncorrelated network, on the other hand, we discover that only a targeted edge-removal attack can induce such correlations.

Generalized theory for node disruption in finite-size complex networks.

After a failure or attack the structure of a complex network changes due to node removal. Here, we show that the degree distribution of the distorted network, under any node disturbances, can be easily computed through a simple formula. Based on this expression, we derive a general condition for the stability of noncorrelated finite complex networks under any arbitrary attack. We apply this formalism to derive an expression for the percolation threshold f_{c} under a general attack of the form f_{k} approximately k;{gamma} , where f_{k} stands for the probability of a node of degree k of being removed during the attack. We show that f_{c} of a finite network of size N exhibits an additive correction which scales as N;{-1} with respect to the classical result for infinite networks.