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Clique percolation in random graphs.
- Source :
-
Physical review. E, Statistical, nonlinear, and soft matter physics [Phys Rev E Stat Nonlin Soft Matter Phys] 2015 Oct; Vol. 92 (4), pp. 042116. Date of Electronic Publication: 2015 Oct 07. - Publication Year :
- 2015
-
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 that they share at least l<k vertices. In this paper we develop a theoretical approach to study clique percolation in Erdős-Rényi graphs, which gives not only the exact solutions of the critical point, but also the corresponding order parameter. Based on this, we prove theoretically that the fraction ψ of cliques in the giant clique cluster always makes a continuous phase transition as the classical percolation. However, the fraction ϕ of vertices in the giant clique cluster for l>1 makes a step-function-like discontinuous phase transition in the thermodynamic limit and a continuous phase transition for l=1. More interesting, our analysis shows that at the critical point, the order parameter ϕ(c) for l>1 is neither 0 nor 1, but a constant depending on k and l. All these theoretical findings are in agreement with the simulation results, which give theoretical support and clarification for previous simulation studies of clique percolation.
Details
- Language :
- English
- ISSN :
- 1550-2376
- Volume :
- 92
- Issue :
- 4
- Database :
- MEDLINE
- Journal :
- Physical review. E, Statistical, nonlinear, and soft matter physics
- Publication Type :
- Academic Journal
- Accession number :
- 26565177
- Full Text :
- https://doi.org/10.1103/PhysRevE.92.042116