  Percolation theory is an approach to study vulnerability of a system. We
develop analytical framework and analyze percolation properties of a network
composed of interdependent networks (NetONet). Typically, percolation of a
single network shows that the damage in the network due to a failure is a
continuous function of the fraction of failed nodes. In sharp contrast, in
NetONet, due to the cascading failures, the percolation transition may be
discontinuous and even a single node failure may lead to abrupt collapse of the
system. We demonstrate our general framework for a NetONet composed of $n$
classic Erd\H{o}s-R\'{e}nyi (ER) networks, where each network depends on the
same number $m$ of other networks, i.e., a random regular network of
interdependent ER networks. In contrast to a \emph{treelike} NetONet in which
the size of the largest connected cluster (mutual component) depends on $n$,
the loops in the RR NetONet cause the largest connected cluster to depend only
on $m$. We also analyzed the extremely vulnerable feedback condition of
coupling. In the case of ER networks, the NetONet only exhibits two phases, a
second order phase transition and collapse, and there is no first phase
transition regime unlike the no feedback condition. In the case of NetONet
composed of RR networks, there exists a first order phase transition when $q$
is large and second order phase transition when $q$ is small. Our results can
help in designing robust interdependent systems.
