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 [Formula: see text] sharing the same set of [Formula: see text] nodes (a two-layer network) whose respective degree distributions [Formula: see text] are given. Defining the overlap coefficient [Formula: see text] as the Jaccard index, we prove that [Formula: see text] is very close to 0 when [Formula: see text] and [Formula: see text] are random and independently generated. We derive an upper bound [Formula: see text] for the maximum overlap coefficient permitted in terms of [Formula: see text] , and [Formula: see text]. Then we present an algorithm based on cross rewiring of links to obtain a two-layer network with any prescribed [Formula: see text] inside the range [Formula: see text]. A refined version of the algorithm allows us to minimize the cross-layer correlations that unavoidably appear for values of [Formula: see text] beyond a critical overlap [Formula: see text]. 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.