Tuning the overlap and the cross-layer correlations in two-layer networks: Application to a susceptible-infectious-recovered 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 α as the Jaccard index, we prove that α is very close to 0 when A and B are random and independently generated. We derive an upper bound α_{M} for the maximum overlap coefficient permitted in terms of p_{A}(k), p_{B}(k), and N. Then we present an algorithm based on cross rewiring of links to obtain a two-layer network with any prescribed α inside the range (0,α_{M}). A refined version of the algorithm allows us to minimize the cross-layer correlations that unavoidably appear for values of α beyond a critical overlap α_{c}<α_{M}. Finally, we present a very simple example of a susceptible-infectious-recovered epidemic model with information dissemination and use the algorithms to determine the impact of the overlap on the final outbreak size predicted by the model.