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…