Robustness of a Network of Networks

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Abstract
Network research has been focused on studying the properties of a single isolated network, which rarely exists. We develop a general analytical framework for studying percolation of n interdependent networks. We illustrate our analytical solutions for three examples: (i) For any tree of n fully dependent Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi (ER) networks, each of average degree k\ifmmode\bar\else\textasciimacron\fi{}, we find that the giant component is P$\infty${}=p[1-exp{}(-k\ifmmode\bar\else\textasciimacron\fi{}P$\infty${})]n where 1-p is the initial fraction of removed nodes. This general result coincides for n=1 with the known second-order phase transition for a single network. For any n>1 cascading failures occur and the percolation becomes an abrupt first-order transition. (ii) For a starlike network of n partially interdependent ER networks, P$\infty${} depends also on the topology\char22{}in contrast to case (i). (iii) For a looplike network formed by n partially dependent ER networks, P$\infty${} is independent of n.
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