Abstract
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.
Highlights
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 each module undergoes a second percolation transition. (ii) When there are more modules with many interconnections between them, the system undergoes a single transition
The fraction of unremoved interconnected nodes, r, is related to the overall fraction of unremoved nodes, p, by r where pinter is the probability that a node is not interconnected. We study this model on both treelike networks of networks (NoNs) and looplike NoNs and find that there are two distinct regimes depending on the number of modules, m
Our results show that modular NoNs can behave significantly differently from random NoNs or spatially embedded NoNs in that they may undergo two separate percolation phase transitions
Summary
Content from this work Abstract may be used under the Many infrastructure networks have a modular structure and are interdependent with other terms of the Creative Commons Attribution 3.0 infrastructures. Attribution to the author(s) and the title of for simplicity, on the case where each network has the same number of communities and the the work, journal citation dependency links are restricted to be between pairs of communities of different networks. 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. 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
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