Creating and controlling overlap in two-layer networks. Application to a mean-field SIS epidemic model with awareness dissemination

We study the properties of the potential overlap between two networks A,B sharing the same set of N nodes (a two-layer network) whose respective degree distributions pA(k), pB(k) are given. Defining the overlap coefficient α as the Jaccard index, we derive upper bounds for the minimum and maximum overlap coefficient in terms of pA(k), pB(k) and N . We also present an algorithm based on cross-rewiring of links to obtain a two-layer network with any prescribed α inside the permitted range. Finally, to illustrate the importance of the overlap for the dynamics of interacting contagious processes, we derive a mean-field model for the spread of an SIS epidemic with awareness against infection over a two-layer network, containing α as a parameter. A simple analytical relationship between α and the basic reproduction number follows. Stochastic simulations are presented to assess the accuracy of the upper bounds of α and the predictions of the mean-field epidemic model.