  Almost all network research has been focused on the properties of a single
network that does not interact and depends on other networks. In reality, many
real-world networks interact with other networks. Here we develop an analytical
framework for studying interacting networks and present an exact percolation
law for a network of $n$ interdependent networks. In particular, we find that
for $n$ Erd\H{o}s-R\'{e}nyi networks each of average degree $k$, the giant
component, $P_{\infty}$, is given by $P_{\infty}=p[1-\exp(-kP_{\infty})]^n$
where $1-p$ is the initial fraction of removed nodes. Our general result
coincides for $n=1$ with the known Erd\H{o}s-R\'{e}nyi second-order phase
transition for a single network. For any $n \geq 2$ cascading failures occur
and the transition becomes a first-order percolation transition. The new law
for $P_{\infty}$ shows that percolation theory that is extensively studied in
physics and mathematics is a limiting case ($n=1$) of a more general general
and different percolation law for interdependent networks.
