Com_cliquePercolation - 02(2005 PHYSICAL REVIEW LETTERS...

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Clique Percolation in Random Networks Imre Dere ´nyi, 1 Gergely Palla, 2 and Tama ´s Vicsek 1,2 1 Department of Biological Physics, Eo ¨tvo ¨s University, Pa ´zma ´ny P. stny. 1A, H-1117 Budapest, Hungary 2 Biological Physics Research Group of HAS, Pa ´zma ´ny P. stny. 1A, H-1117 Budapest, Hungary (Received 10 November 2004; published 29 April 2005) The notion of k -clique percolation in random graphs is introduced, where k is the size of the complete subgraphs whose large scale organizations are analytically and numerically investigated. For the Erdo ˝s- Re ´nyi graph of N vertices we obtain that the percolation transition of k -cliques takes place when the probability of two vertices being connected by an edge reaches the threshold p c ± k ²³´ ± k ÿ 1 ² N µ ÿ 1 = ± k ÿ 1 ² . At the transition point the scaling of the giant component with N is highly nontrivial and depends on k .We discuss why clique percolation is a novel and ef±cient approach to the identi±cation of overlapping communities in large real networks. DOI: 10.1103/PhysRevLett.94.160202 PACS numbers: 02.10.Ox, 05.70.Fh, 64.60. 2 i, 89.75.Hc There has been a quickly growing interest in networks, since they can represent the structure of a wide class of complex systems occurring from the level of cells to society. Data obtained on real networks show that the corresponding graphs exhibit unexpected nontrivial prop- erties, e.g., anomalous degree distributions, diameter, spreading phenomena, clustering coef±cient, and correla- tions [1]. Very recently great attention has been paid to the local structural units of networks. Small and well de±ned subgraphs have been introduced as ‘‘motifs’’ [2]. Their distribution and clustering properties [2,3] can be used to interpret global features as well. Somewhat larger units, made up of vertices that are more densely connected to each other than to the rest of the network, are often referred to as communities [4], and have been considered to be the essential structural units of real networks. They have no obvious de±nition, and most of the recent methods for their identi±cation rely on dividing the network into smaller pieces. The biggest drawback of these methods is that they do not allow for overlapping communities, although overlaps are generally assumed to be crucial features of communities. In this Letter we lay down the fundamentals of a kind of percolation phenomenon on graphs, which can also be used as an effective and deterministic method for uniquely identifying overlapping communities in large real networks [5]. Meanwhile, the various aspects of the classical Erdo Re ´nyi (ER) uncorrelated random graph [6] remain still of great interest since such a graph can serve both as a test bed for checking all sorts of new ideas concerning complex networks in general, and as a prototype to which all other random graphs can be compared. Perhaps the most con- spicuous early result on the ER graphs was related to the percolation transition taking place at p ³ p c 1 =N , where p is the probability that two vertices are connected
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Com_cliquePercolation - 02(2005 PHYSICAL REVIEW LETTERS...

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