It is well known that real-world networks present correlations among vertices. Such correlations
between degrees of vertices in a network essentially characterize its structure.  For instance, social
networks show assortative mixing, i.e., a preference of high-degree vertices to be connected to other
high-degree vertices, while technological and biological networks are mostly disassortative, i.e., their
high-degree vertices tend to be connected to low-degree ones.
I will present an analytical study of the problem of percolation on an equilibrium random network
with degree-degree correlations between nearest-neighboring vertices focusing on critical
singularities at a percolation threshold. I will show the criteria for degree-degree correlations to be
irrelevant for critical singularities. I will also present examples of networks in which assortative and
disassortative mixing leads to unusual percolation properties and new critical exponents.

Goltsev, A. V.; Dorogovtsev, S. N.; Mendes, J. F. F., Phys. Rev. E 78, 051105 (2008)