  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.
