Resilience of Networks Formed of Interdependent Modular Networks

Abstract

Many infrastructure networks have a modular structure and are also interdependent. 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 by simulations. We focus 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 very realistic for infrastructure across cities. Each city has its own infrastructures and different infrastructures are dependent 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 been shown that single networks are very susceptible to the failure of the interconnected nodes (between communities) Shai et al. and that attacks on these nodes are more crippling than attacks based on betweenness da Cunha et al. 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 influence the type of transitions observed in interdependent networks and should be considered in attempts to make interdependent networks more resilient.

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• Network Properties for Robust Multilayer Infrastructure Systems: A Percolation Theory Review
• Scaling of percolation transitions on Erdös-Rényi networks under centrality-based attacks
• Resilience of spatial networks with inter-links behaving as an external field
• Percolation of Hierarchical Networks and Networks of Networks
• Spreading of localized attacks on spatial multiplex networks with a community structure
• The structure of networks that evolve under a combination of growth, via node addition and random attachment, and contraction, via random node deletion
• Two transitions in spatial modular networks
• Critical field-exponents for secure message-passing in modular networks
• Dual communities in spatial networks
• Performance of attack strategies on modular networks
• Message-Passing on Hypergraphs: Detectability, Phase Transitions and Higher-Order Information
• Phase transition in evolving networks that combine preferential attachment and random node deletion
• Optimization of robustness based on reinforced nodes in a modular network
• Cascading failures in anisotropic interdependent networks of spatial modular structures
• Empirical determination of the optimum attack for fragmentation of modular networks