## Abstract

Complex networks appear in almost every aspect of science and technology. Previous work in network theory has focused primarily on analyzing single networks that do not interact with other networks, despite the fact that many real-world networks interact with and depend on each other. Very recently an analytical framework for studying the percolation properties of interacting networks has been introduced. Here we review the analytical framework and the results for percolation laws for a network of networks (NON) formed by interdependent random networks. The percolation properties of a network of networks differ greatly from those of single isolated networks. In particular, although networks with broad degree distributions, e.g., scale-free networks, are robust when analyzed as single networks, they become vulnerable in a NON. Moreover, because the constituent networks of a NON are connected by node dependencies, a NON is subject to cascading failure. When there is strong interdependent coupling between networks, the percolation transition is discontinuous (is a first-order transition), unlike the well-known continuous second-order transition in single isolated networks. We also review some possible real-world applications of NON theory.

Original language | English |
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Title of host publication | Networks of Networks |

Subtitle of host publication | The Last Frontier of Complexity |

Publisher | Springer Verlag |

Pages | 3-36 |

Number of pages | 34 |

ISBN (Print) | 9783319035178 |

DOIs | |

State | Published - 2014 |

### Publication series

Name | Understanding Complex Systems |
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ISSN (Print) | 1860-0832 |

ISSN (Electronic) | 1860-0840 |

### Bibliographical note

Funding Information:We wish to thank ONR (Grant N00014-09-1-0380, Grant N00014-12-1-0548), DTRA (Grant HDTRA-1-10-1- 0014, Grant HDTRA-1-09-1-0035), NSF (Grant CMMI 1125290), the European EPIWORK, MULTIPLEX, CONGAS (Grant FP7-ICT-2011-8-317672), FET Open Project FOC 255987 and FOC-INCO 297149, and LINC projects, DFG, the Next Generation Infrastructure (Bsik) and the Israel Science Foundation for financial support. SVB acknowledges the Dr. Bernard W. Gamson Computational Science Center at Yeshiva College.