Robustness of a Network of Networks
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 $\overline{k}$, we find that the giant component is ${P}_{\ensuremath{\infty}}=p[1\ensuremath{-}\mathrm{exp} (\ensuremath{-}\overline{k}{P}_{\ensuremath{\infty}}){]}^{n}$ where $1\ensuremath{-}p$ 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, ${P}_{\ensuremath{\infty}}$ depends also on the topology---in contrast to case (i). (iii) For a looplike network formed by $n$ partially dependent ER networks, ${P}_{\ensuremath{\infty}}$ is independent of $n$.