Catastrophic cascade of failures in interdependent networks

Abstract

Complex networks have been studied intensively for a decade, but research still focuses on the limited case of a single, non-interacting network. Modern systems are coupled together and therefore should be modelled as interdependent networks. A fundamental property of interdependent networks is that failure of nodes in one network may lead to failure of dependent nodes in other networks. This may happen recursively and can lead to a cascade of failures. In fact, a failure of a very small fraction of nodes in one network may lead to the complete fragmentation of a system of several interdependent networks. A dramatic real-world example of a cascade of failures ('concurrent malfunction') is the electrical blackout that affected much of Italy on 28 September 2003: the shutdown of power stations directly led to the failure of nodes in the Internet communication network, which in turn caused further breakdown of power stations. Here we develop a framework for understanding the robustness of interacting networks subject to such cascading failures. We present exact analytical solutions for the critical fraction of nodes that, on removal, will lead to a failure cascade and to a complete fragmentation of two interdependent networks. Surprisingly, a broader degree distribution increases the vulnerability of interdependent networks to random failure, which is opposite to how a single network behaves. Our findings highlight the need to consider interdependent network properties in designing robust networks.

Networks formed from interdependent networks

“…For a fully interdependent case in which each node in one network depends on a functioning node in other networks and vice versa, a first-order discontinuous phase transition, which is dramatically different from the second-order continuous phase transition found in isolated networks ( Fig. 1), was found 73 . This phenomenon is caused by the presence of two types of link: connectivity links within each network; and dependence links between networks.…”

“…Recently, motivated by the fact that modern, crucially important infrastructures significantly interact, a mathematical framework was developed 73 to study percolation in a system of two interdependent networks subject to cascading failure. The analytical framework is based on a generating-function formalism widely used for studies of percolation and structure within a single network [73][74][75] .…”

“…The analytical framework is based on a generating-function formalism widely used for studies of percolation and structure within a single network [73][74][75] . The framework for interdependent networks enables us to follow the dynamics of the cascading failures as well as to derive the analytic solutions for the final steady state.…”

“…The framework for interdependent networks enables us to follow the dynamics of the cascading failures as well as to derive the analytic solutions for the final steady state. It was found 73 that certain types of interdependent network were significantly more vulnerable than their non-interacting counterparts. The failure of even a small number of elements within a single network may trigger a catastrophic cascade of events that destroys the global connectivity.…”

“…In all of the above studies 73,[76][77][78] , the dependent pairs of nodes in both networks were chosen randomly. Thus when highdegree nodes in one network depend with a high probability on low-degree nodes of another network the configuration becomes vulnerable.…”