Percolation of hierarchical networks and networks of networks

Much work has been devoted to studying percolation of networks and interdependent networks under varying levels of failures. Researchers have considered many different realistic network structures such as scale-free networks, spatial networks, and more. However, thus far no study has analyzed a system of hierarchical community structure of many networks. For example, infrastructure across cities is likely distributed with nodes tightly connected within small neighborhoods, somewhat less connected across the whole city, and even fewer connections between cities. Furthermore, while previous work identified interconnected nodes, those nodes with links outside their neighborhood, to be more likely to be attacked or to fail, in a hierarchical structure nodes can be interconnected in different layers (between neighborhoods, between cities, etc.). We consider here the case where the nodes with interconnections at the highest level of the hierarchy are most likely to fail, followed by those with interconnections at the next level, etc. This is because nodes at higher levels of the hierarchy have the longest links as well as having more flow passing through them. We develop an analytic solution for percolation of both single and interdependent networks of this structure and verify our theory through simulations. We find that, depending on the number of levels in the hierarchy, there may be multiple transitions in the giant component (fraction of connected nodes), as the network separates at the various levels. Our results show that these multiple jumps are a feature of hierarchical networks and can affect the vulnerability of infrastructure networks.