### Abstract

Motivated by the success of a k-clique percolation method for the identification of overlapping communities in large real networks, here we study the k-clique percolation problem in the Erdos-Rényi graph. When the probability p of two nodes being connected is above a certain threshold p _{c} (k), the complete subgraphs of size k (the k-cliques) are organized into a giant cluster. By making some assumptions that are expected to be valid below the threshold, we determine the average size of the k-clique percolation clusters, using a generating function formalism. From the divergence of this average size we then derive an analytic expression for the critical linking probability p _{c} (k).

Original language | English |
---|---|

Pages (from-to) | 219-227 |

Number of pages | 9 |

Journal | Journal of Statistical Physics |

Volume | 128 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Jul 2007 |

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### Keywords

- Community
- Network
- Percolation
- Random graph

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**The critical point of k-clique percolation in the Erdos-Rényi graph.** / Palla, Gergely; Derényi, I.; Vicsek, T.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 128, no. 1-2, pp. 219-227. https://doi.org/10.1007/s10955-006-9184-x

}

TY - JOUR

T1 - The critical point of k-clique percolation in the Erdos-Rényi graph

AU - Palla, Gergely

AU - Derényi, I.

AU - Vicsek, T.

PY - 2007/7

Y1 - 2007/7

N2 - Motivated by the success of a k-clique percolation method for the identification of overlapping communities in large real networks, here we study the k-clique percolation problem in the Erdos-Rényi graph. When the probability p of two nodes being connected is above a certain threshold p c (k), the complete subgraphs of size k (the k-cliques) are organized into a giant cluster. By making some assumptions that are expected to be valid below the threshold, we determine the average size of the k-clique percolation clusters, using a generating function formalism. From the divergence of this average size we then derive an analytic expression for the critical linking probability p c (k).

AB - Motivated by the success of a k-clique percolation method for the identification of overlapping communities in large real networks, here we study the k-clique percolation problem in the Erdos-Rényi graph. When the probability p of two nodes being connected is above a certain threshold p c (k), the complete subgraphs of size k (the k-cliques) are organized into a giant cluster. By making some assumptions that are expected to be valid below the threshold, we determine the average size of the k-clique percolation clusters, using a generating function formalism. From the divergence of this average size we then derive an analytic expression for the critical linking probability p c (k).

KW - Community

KW - Network

KW - Percolation

KW - Random graph

UR - http://www.scopus.com/inward/record.url?scp=34249875971&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34249875971&partnerID=8YFLogxK

U2 - 10.1007/s10955-006-9184-x

DO - 10.1007/s10955-006-9184-x

M3 - Article

AN - SCOPUS:34249875971

VL - 128

SP - 219

EP - 227

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1-2

ER -