Random graph model with power-law distributed triangle subgraphs

Details Random graph model with power-law distributed triangle subgraphs Random graph model with power-law distributed triangle subgraphs

2004-12-17

arxiv.org/abs/cond-mat/0412472v2

Physics

Condensed Matter

Disordered Systems and Neural Networks

4 pages, 2 figure; v2: major conceptual changes

Scientific paper

10.1103/PhysRevE.72.025103

Clustering is well-known to play a prominent role in the description and understanding of complex networks, and a large spectrum of tools and ideas have been introduced to this end. In particular, it has been recognized that the abundance of small subgraphs is important. Here, we study the arrangement of triangles in a model for scale-free random graphs and determine the asymptotic behavior of the clustering coefficient, the average number of triangles, as well as the number of triangles attached to the vertex of maximum degree. We prove that triangles are power-law distributed among vertices and characterized by both vertex and edge coagulation when the degree exponent satisfies $2<\beta<2.5$; furthermore, a finite density of triangles appears as $\beta=2+1/3$.

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