Exact solutions for social and biological contagion models on mixed   directed and undirected, degree-correlated random networks
  We derive analytic expressions for the possibility, probability, and expected size of global spreading events starting from a single infected seed for a broad collection of contagion processes acting on random networks with both directed and undirected edges and arbitrary degree-degree correlations. Our work extends previous theoretical developments for the undirected case, and we provide numerical support for our findings by investigating an example class of networks for which we are able to obtain closed-form expressions. 
Joshua L. Payne, Kameron Decker Harris, and Peter Sheridan Dodds
physics.soc-ph cond-mat.dis-nn cs.SI