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 $p_A(k), p_B(k)$ are given. Defining the overlap coefficient $\alpha$ as the Jaccard index, we derive upper bounds for the minimum and maximum overlap coefficient in terms of $p_A(k)$, $p_B(k)$ and $N$. We also present an algorithm based on cross-rewiring of links to obtain a two-layer network with any prescribed $\alpha$ 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 $\alpha$ as a parameter. A simple analytical relationship between $\alpha$ and the basic reproduction number follows. Stochastic simulations are presented to assess the accuracy of the upper bounds of $\alpha$ and the predictions of the mean-field epidemic model.