Percolation transition in networks with degree-degree correlation.

We introduce an exponential random graph model for networks with a fixed degree distribution and a tunable degree-degree correlation. We then investigate the nature of the percolation transition in a correlated network with a Poisson degree distribution. It is found that negative correlation is irrelevant in that the percolation transition in the disassortative network belongs to the same universality class as in the uncorrelated network. Positive correlation turns out to be relevant. The percolation transition in the assortative network is characterized by the nondiverging mean size of finite clusters and power-law scalings of the density of the largest cluster and the cluster size distribution in the nonpercolating phase as well as at the critical point. Our results suggest that the unusual type of percolation transition in the growing network models reported recently may be inherited from the assortative degree-degree correlation.