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)similar to ky, 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.