@article{gao2013percolation,
author = "Jianxi Gao and Sergey V. Buldyrev and H. Eugene Stanley and Xiaoming Xu and Shlomo Havlin",
abstract = "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$\backslash$H\{o\}s-R$\backslash$'\{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 $\backslash$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.",
doi = "10.1103/PhysRevE.88.062816",
issn = "15393755",
journal = "Physical Review E - Statistical, Nonlinear, and Soft Matter Physics",
number = "6",
pages = "1--13",
title = "{P}ercolation of a general network of networks",
volume = "88",
year = "2013",
}