Abrupt transition in the structural formation of interconnected networks

Real-world networks are rarely isolated. A model of an interdependent network of networks shows that an abrupt phase transition occurs when interconnections between independent networks are added. This study also suggests ways to minimize the danger of abrupt structural changes to real networks. Our world is linked by a complex mesh of networks through which information, people and goods flow. These networks are interdependent on each other, and present structural and dynamical features1,2,3,4,5,6 different from those observed in isolated networks7,8,9. Although examples of such dissimilar properties are becoming more abundant—such as in diffusion, robustness and competition—it is not yet clear where these differences are rooted. Here we show that the process of building independent networks into an interconnected network of networks undergoes a structurally sharp transition as the interconnections are formed. Depending on the relative importance of inter- and intra- layer connections, we find that the entire interdependent system can be tuned between two regimes: in one regime, the various layers are structurally decoupled and they act as independent entities; in the other regime, network layers are indistinguishable and the whole system behaves as a single-level network. We analytically show that the transition between the two regimes is discontinuous even for finite-size networks. Thus, any real-world interconnected system is potentially at risk of abrupt changes in its structure, which may manifest new dynamical properties.