Correlated Edge Overlaps in Multiplex Networks
  We develop the theory of sparse multiplex networks with partially overlapping links based on their local tree-likeness. This theory enables us to find the giant mutually connected component in a two-layer multiplex network with arbitrary correlations between connections of different types. We find that correlations between the overlapping and non-overlapping links markedly change the phase diagram of the system, leading to multiple hybrid phase transitions. For assortative correlations we observe recurrent hybrid phase transitions. 
Gareth J. Baxter, Ginestra Bianconi, Rui A. da Costa, Sergey N.   Dorogovtsev, Jos\'e F. F. Mendes
physics.soc-ph cond-mat.dis-nn