In recent years, due to the advances in technology, many systems have become more and more integrated and interdependent. This dependence between systems can cause to a spread of damages leading to a cascade of failures and abrupt collapse. Therefore, many studies have been carried out to analyze cascading failures in interdependent networks [1]. In particular, recent studies show that due to dependencies, localized attacks on homogeneous spatial interdependent systems are significantly more damaging than random attacks. In addition, many real networks have a modular structure, such as biological networks and many infrastructure systems.
Our study combines for the first time, three ubiquitous features of real complex systems --- interdependence, spatiality and modularity. We study the effect of localized attacks on a multiplex spatial network, where each layer is a network of communities [2].
We model the multiplex composed of two layers in which the nodes are placed at sites of a square lattice. Each layer is constructed as ER networks (representing communities) that are tiled and connected to each other as a square lattice. Thus, the links within a community (intra-links) are connected at random, while the links between the communities (inter-links) can only connect between neighboring tiles. This model can represent an interdependent infrastructure system of cities where within the city there are many links while between cities there are fewer links.
We find, both by simulation and theory, that for different parameters of connectivity and spatiality --- there is a critical localized size of damage above which the damage spreads and the system collapses. In addition, we find that for the same number of total links, the networks with low interconnectivity are more robust against localized attacks than the system in which the communities are well connected. These results can help improving the resilience of realistic infrastructures.

Refs:
[1] S V Buldyrev et al. Catastrophic cascade of failures in interdependent networks. Nature 464, 08932 (2010).
[2] D Vaknin el al. Spreading of localized attacks on spatial multiplex networks with a community structure. arXiv:1912.05677.