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Many real systems such as, roads, shipping routes, and infrastructure systems can be modeled based on spatially embedded networks. The inter-links between two distant spatial networks, such as those formed by transcontinental airline flights, play a crucial role in optimizing communication and transportation over such long distances. Still, little is known about how inter-links affect the structural resilience of such systems. Here, we develop a framework to study the structural resilience of interlinked spatially embedded networks based on percolation theory. We find that the inter-links can be regarded as an external field near the percolation phase transition, analogous to a magnetic field in a ferromagnetic–paramagnetic spin system. By defining the analogous critical exponents δ and γ, we find that their values for various inter-links structures follow Widom's scaling relations. Furthermore, we study the optimal robustness of our model and compare it with the analysis of real-world networks. The framework presented here not only facilitates the understanding of phase transitions with external fields in complex networks but also provides insight into optimizing real-world infrastructure networks.

Many infrastructure networks have a modular structure and are also interdependent with other infrastructures. While significant research has explored the resilience of interdependent networks, there has been no analysis of the effects of modularity. Here we develop a theoretical framework for attacks on interdependent modular networks and support our results through simulations. We focus, for simplicity, on the case where each network has the same number of communities and the dependency links are restricted to be between pairs of communities of different networks. This is particularly realistic for modeling infrastructure across cities. Each city has its own infrastructures and different infrastructures are dependent only within the city. However, each infrastructure is connected within and between cities. For example, a power grid will connect many cities as will a communication network, yet a power station and communication tower that are interdependent will likely be in the same city. It has previously been shown that single networks are very susceptible to the failure of the interconnected nodes (between communities) (Shai et al 2014 arXiv:1404.4748) and that attacks on these nodes are even more crippling than attacks based on betweenness (da Cunha et al 2015 arXiv:1502.00353). In our example of cities these nodes have long range links which are more likely to fail. For both treelike and looplike interdependent modular networks we find distinct regimes depending on the number of modules, m. (i) In the case where there are fewer modules with strong intraconnections, the system first separates into modules in an abrupt first-order transition and then each module undergoes a second percolation transition. (ii) When there are more modules with many interconnections between them, the system undergoes a single transition. Overall, we find that modular structure can significantly influence the type of transitions observed in interdependent networks and should be considered in attempts to make interdependent networks more resilient.