Convergence towards an Erdős-Rényi graph structure in network contraction processes.

Abstract

In a highly influential paper twenty years ago, Barabási and Albert [Science 286, 509 (1999)SCIEAS0036-807510.1126/science.286.5439.509] showed that networks undergoing generic growth processes with preferential attachment evolve towards scale-free structures. In any finite system, the growth eventually stalls and is likely to be followed by a phase of network contraction due to node failures, attacks, or epidemics. Using the master equation formulation and computer simulations, we analyze the structural evolution of networks subjected to contraction processes via random, preferential, and propagating node deletions. We show that the contracting networks converge towards an Erdős-Rényi network structure whose mean degree continues to decrease as the contraction proceeds. This is manifested by the convergence of the degree distribution towards a Poisson distribution and the loss of degree-degree correlations.