Interdependent networks

In an attempt to provide a mathematical framework for analysing the consequences of cascades of failures occurring in interdependent critical infrastructures [194, 195], Buldyrevet al.[53] defined a new class of networks calledinterdependent networks, and studied a percolation process in a system of two interdependent networks. In these networks, unlike the connectivity links within each networks, the networks are interconnected bydependency links, representing the fact that the function of a given node in one network depends crucially on nodes in other networks. Thus, when a node from one network is removed in a percolation process, its dependent node from the other network is automatically removed as well.

This model was designed to capture the situation observed in real-world data from a power network and an Internet network (a supervisory control and data acquisition system). The data was extracted from an electrical blackout that affected much of Italy in 2003, where the shutdown of power stations directly led to the failure of nodes in the Internet communication network (since switches rely on electricity), which in turn caused further breakdown of power stations (since they rely on the Internet for control and recovery) [195].

In their model, Buldyrevet al. consider two equal size networks with arbitrary degree distribution, where each node from one network is mutually dependent on a randomly selected node from the other network. Using a generating function approach, they show that the resulting networks

are significantly more vulnerable than their non-interacting counterparts, where small failure in one network may lead to catastrophic consequences that breaks the whole system. This behavior is characteristic of a first-order phase transition, in contrast to the second-order phase transition characterising percolation of a single network, where the size of the giant component is decreasingcontinuouslyas the number of removed nodes increases, see Fig. 2.6. Here, instead, when a critical number of nodes are removed, the giant component suddenly (i.e. discontinuously) collapses, resulting in a first order percolation transition. Perhaps more importantly that the qualitative change in the percolation transition, the authors show that in contrast to single networks, interdependent networks with broader degree distributions are more vulnerable. Specifically, the result where single scale-free networks always percolate in the limit of large networks (see section 2.3.1), is no longer valid for interdependent scale-free networks.

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Figure 2.6:Second order vs. first order percolation transition. (a) In two weakly interdependent networks (with few interdependency links between them), the size (fraction of nodes) of the largest connected component is changing continuously with the fraction of occupied nodes in the network. However,

in strongly interdependent networks, the transition is discontinuous “jumping” from 0 to≈0.4 at the

percolation threshold. In other words, the giant component suddenly forms in this case, where in weakly

interdependent networks it forms gradually. (b)nfully interdependent networks (all nodes mutually

depend on nodes in other networks). As the number of networksnincreases, the “jump” in the size of the

largest connected component is bigger. Figure adapted from [103].

As stated earlier, the model of Buldyrev et al. has attracted considerable attention and consequently has been extended in various ways [103]. For example, in a system of two interdependent ER networks where only part of the nodes are interdependent (called partially

instead of fully interdependent networks), there exists a critical fraction of interdependent nodes above which the system is very fragile, exhibiting a first order percolation transition [180]. In case the networks are scale-free, a very interesting “hybrid” phase between first and second order was found, where the size of the giant component has a sharp drop from finite value to a much smaller, yet a nonzero value [249]. This “hybrid” transition was also found in a system of two ER networks interconnected by both connectivity and dependency links [127].

A theoretical framework for understanding the robustness of interdependent networks under targeted attacks on specific degree nodes was developed in [130], and it was found that even when hubs have low failure probability (i.e. nodes are attacked in order from lowest degree to highest degree), interdependent scale-free networks have percolation threshold much larger than zero. This is because high-degree nodes often depend on the more common low-degree nodes in the other network. And indeed, interdependent networks where nodes depend on similar- degree nodes (i.e. when the inter-degrees are positively correlated), called intersimilar networks, are much more robust [54, 128, 179]. However, degree correlation within a single network (i.e., the likelihood of nodes with similar degree to be connected), as well as high clustering coefficient, has been shown to decrease the robustness of the entire system [131, 206, 239, 250]. Recently, there has been several efforts to extend the theory of interdependent networks from two networks to a network of interdependent networks (NON) [40, 41, 101, 102]. Also, recent papers examine design strategies for mitigating risk in systems of interdependent networks, which can be done at a very low cost with careful selection of autonomous nodes (nodes which are not interdependent) [202, 229].

Finally, since one of the main motivation for the work of Buldyrev et al. was to address catastrophic cascade of events in critical infrastructures, and these are often spatially embedded (e.g. power grids, Internet), several models of interdependent networks were developed taking spatial constraints into account. For example, it has been shown that in a system composed of two interdependent square lattice networks placed on the same Cartesian plane, there exist a critical length of dependency links (i.e. the maximum distance between a node in one network and the node that it depends on in another network),r_c, above which the system is very fragile, characterized by an abrupt collapse of the giant component (first order transition) [75, 148]. In a system ofninterdependent networks,r_chas been shown to rapidly decrease withn[209]. This result was also verified in a system of two identical random regular graphs (instead of lattices), where networks with long dependency distances (shortest path between a pair of interdependent nodes) are much more vulnerable than networks with short dependency distances [140]. A very strong result was obtained in [30], where interdependent spatially embedded networks with randomly assigned interdependency links (i.e. with no length constraint) were found to be

extremely vulnerable: the failure ofanysmall fraction of interdependent nodes leads to an abrupt collapse. Finally, Berezinet al.[37] studied localised geographical attacks in two interdependent spatially embedded networks and found a critical attack size above which it will spread through the entire system and lead to its complete collapse.