The Critical Point of k-Clique Percolation in the Erdős Rényi Graph
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 Erdős-Rényi graph. When the probability p of two nodes being connected is above a certain threshold pc(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 pc(k).
- Publication:
-
Journal of Statistical Physics
- Pub Date:
- July 2007
- DOI:
- 10.1007/s10955-006-9184-x
- arXiv:
- arXiv:cond-mat/0610298
- Bibcode:
- 2007JSP...128..219P
- Keywords:
-
- random graph;
- network;
- percolation;
- community;
- Condensed Matter - Disordered Systems and Neural Networks;
- Condensed Matter - Statistical Mechanics
- E-Print:
- 12 pages, 2 figures