The robustness of interdependent clustered networks

It was recently found that cascading failures can cause the abrupt breakdown of a system of interdependent networks. Using the percolation method developed for single clustered networks by Newman [Phys. Rev. Lett. 103, 058701 (2009)], we develop an analytical method for studying how clustering within the networks of a system of interdependent networks affects the system’s robustness. We find that clustering significantly increases the vulnerability of the system, which is represented by the increased value of the percolation threshold pc in interdependent networks. Introduction. – In a system of interdependent networks, the functioning of nodes in one network is dependent upon the functioning of nodes in other networks of the system. The failure of nodes in one network can cause nodes in other networks to fail, which in turn can cause further damage to the first network, leading to cascading failures and catastrophic consequences. Power blackouts across entire countries have been caused by cascading failures between the interdependent communication and power grid systems [1, 2]. Because infrastructures in our modern society are becoming increasingly interdependent, understanding how systemic robustness is affected by these interdependencies is essential if we are to design infrastructures that are resilient [3–6]. In addition to research carried out on specific systems [7–16], a mathematical framework [17] and its generalizations [18–20] have been developed recently. These studies use a percolation approach to analyze a system of two or more interdependent networks subject to cascading failure [21, 22]. It was found that interdependent networks are significantly more vulnerable than their stand-alone counterparts. The dynamics of cascading failure are strongly affected by the structure patterns of network components and by the interaction between networks. This research has focused almost exclusively on random interdependent networks in which clustering within component networks is small or approaches zero. Clustering quantifies the propensity for two neighbors of the same vertex to also be neighbors of each other, forming triangle-shaped configurations in the network [23–25]. Unlike random networks in which there is very little or no clustering, real-world networks exhibit significant clustering. Recent studies have shown that, for single networks, both bond percolation and site percolation in clustered networks have higher epidemic thresholds compared to the unclustered networks [26–31]. Here we present a mathematical framework for understanding how the robustness of interdependent networks is affected by clustering within the network components. We extend the percolation method developed by Newman [26] for single clustered networks to coupled clustered networks. We find that interdependent networks that exhibit significant clustering are more vulnerable to random node failure than networks without significant clustering. We are able to simplify our interdependent networks model— without losing its general applicability—by reducing its size to two networks, A and B, each having the same number of nodes N . The N nodes in A and B have bidirectional dependency links to each other, establishing a oneto-one correspondence. Thus the functioning of a node in network A depends on the functioning of the correspond-