Spreading of localized attacks on spatial multiplex networks with a community structure

We study the effect of localized attacks on a multiplex network, where each layer is a network of communities embedded in space. We assume that nodes are densely connected within a community and sparsely connected to the nodes in the neighboring communities. To investigate percolation processes in this realistic system we develop an analytical scheme, applying the finite-element method. We find, both by simulation and theory, that in many cases there is a critical size of localized damage above which it will spread and the entire system will collapse. In addition, we show that for a constant number of links, networks with less connectivity between communities are surprisingly more robust.