Percolation transition in networks with degree-degree correlation

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