{
  "id": "1508.01878",
  "version": "v1",
  "published": "2015-08-08T09:58:07.000Z",
  "updated": "2015-08-08T09:58:07.000Z",
  "title": "Clique percolation in random graphs",
  "authors": [
    "Ming Li",
    "Youjin Deng",
    "Bing-Hong Wang"
  ],
  "comment": "5 pages, 5 figures",
  "categories": [
    "cond-mat.stat-mech",
    "cs.SI",
    "physics.soc-ph"
  ],
  "abstract": "As a generation of the classical percolation, clique percolation focuses on the connection of cliques in a graph, where the connection of two $k$-cliques means they share at leat $l<k$ vertices. In this paper, we develop a theoretical approach to study clique percolation in Erd\\H{o}s-R\\'enyi graphs, which can give the exact solutions of the order parameter and the critical point. We find that the fraction of cliques in the giant clique cluster always takes a continuous phase transition as the classical percolation. However, the fraction of vertices in the giant clique cluster demonstrates an unnormal phase transitions for $k>3$ and $l>1$, i.e., the fraction of vertices in the giant clique cluster always takes value $1$ above the critical point, however, a size-independent constant at the critical point. Together with the analysis of the finite size scaling and cluster number distribution, we give a theoretical support and clarification for previous simulation studies of the clique percolation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-08T09:58:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random graphs",
      "critical point",
      "giant clique cluster demonstrates",
      "clique percolation focuses",
      "unnormal phase transitions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}