## Abstract

Percolation theory is extensively studied in statistical physics and mathematics with applications in diverse fields. However, the research is focused on systems with only one type of links, connectivity links. We review a recently developed mathematical framework for analyzing percolation properties of realistic scenarios of networks having links of two types, connectivity and dependency links. This formalism was applied to study Erdo{double acute}s-Rényi (ER) networks that include also dependency links. For an ER network with average degree k̄ that is composed of dependency clusters of size s, the fraction of nodes that belong to the giant component, P_{∞}, is given by P_{∞}=p^{s-1}[1-exp(-k̄pP_{∞})]^{s} where 1-p is the initial fraction of randomly removed nodes. Here, we apply the formalism to the study of random-regular (RR) networks and find a formula for the size of the giant component in the percolation process: P_{∞}=p^{s-1}(1-r^{k})^{s} where r is the solution of r=p^{s}(r^{k-1}-1)(1-r^{k})+1, and k is the degree of the nodes. These general results coincide, for s=1, with the known equations for percolation in ER and RR networks respectively without dependency links. In contrast to s=1, where the percolation transition is second order, for s>1 it is of first order. Comparing the percolation behavior of ER and RR networks we find a remarkable difference regarding their resilience. We show, analytically and numerically, that in ER networks with low connectivity degree or large dependency clusters, removal of even a finite number (zero fraction) of the infinite network nodes will trigger a cascade of failures that fragments the whole network. Specifically, for any given s there exists a critical degree value, k̄_{min}, such that an ER network with k̄ ≥ k̄_{min} is unstable and collapse when removing even a single node. This result is in contrast to RR networks where such cascades and full fragmentation can be triggered only by removal of a finite fraction of nodes in the network.

Original language | English |
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Pages (from-to) | 686-695 |

Number of pages | 10 |

Journal | Journal of Statistical Physics |

Volume | 145 |

Issue number | 3 |

DOIs | |

State | Published - Nov 2011 |

### Bibliographical note

Funding Information:Acknowledgements We thank Yanqing Hu for helpful discussions. We thank the European EPIWORK project, the Israel Science Foundation, ONR, DFG, and DTRA for financial support.

### Funding

Acknowledgements We thank Yanqing Hu for helpful discussions. We thank the European EPIWORK project, the Israel Science Foundation, ONR, DFG, and DTRA for financial support.

Funders | Funder number |
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Office of Naval Research | |

Defense Threat Reduction Agency | |

Seventh Framework Programme | 231807 |

Deutsche Forschungsgemeinschaft | |

Israel Science Foundation |

## Keywords

- Cascade of failures
- Dependency links
- Networks
- Percolation