  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.
