Percolation, Cascades, and Congruence in Interdependent Networks

Understanding the network structure of individual systems has lead to tremendous advances in the past decade, with applications ranging from spread of disease on networks of human contacts, to collaboration patterns of scientists, to interactions among proteins. A main focus has been on connectivity and resilience of such individual networks. Yet, in reality no network lives in isolation. E-commerce networks rely on the World-Wide-Web, which in turn relies on the physical internet, the power grid, specialized communications networks, and river networks. Gene regulatory networks are coupled fundamentally to protein interaction networks and to metabolic networks. Despite the pervasiveness of interdependent networks, the consequences of interdependence are largely unexplored. Here we show several surprising consequences that result when individual networks are put into the more realistic and larger context of networks of networks. First we show interactions can enhance or delay the onset of large scale connectivity (the so called percolation transition). Second, we consider simple dynamical processes on coupled networks, modeling cascades of load shedding, and show that some level of coupling is beneficial, yet further increases in coupling are ultimately detrimental. Finally we consider congruence and mismatch between networks of different types, such as social networks and the structure of the networks produced by social organizations, with explicit examples from Open Source Software systems.