Multiple structural transitions in interacting networks

Many real-world systems can be modeled as interconnected multilayer networks, namely, a set of networksinteracting with each other. Here, we present a perturbative approach to study the properties of a general class ofinterconnected networks as internetwork interactions are established. We reveal multiple structural transitions forthe algebraic connectivity of such systems, between regimes in which each network layer keeps its independentidentity or drives diffusive processes over the whole system, thus generalizing previous results reporting a singletransition point. Furthermore, we show that, at first order in perturbation theory, the growth of the algebraicconnectivity of each layer depends only on the degree configuration of the interaction network (projected on therespective Fiedler vector), and not on the actual interaction topology. Our findings can have important implicationsin the design of robust interconnected networked systems, particularly in the presence of network layers whoseintegrity is more crucial for the functioning of the entire system. We finally show results of perturbation theoryapplied to the adjacency matrix of the interconnected network, which can be useful to characterize percolationprocesses on such systems.