Random graph model with power-law distributed triangle subgraphs

Abstract

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 $\beta =2+1/3$.

4 pages, 2 figure; v2: major conceptual changes

References in corpus (9)

• The structure and function of complex networks
• Class of correlated random networks with hidden variables
• The topological relationship between the large-scale attributes and local interaction patterns of complex networks
• Subgraphs in random networks
• Clustering of correlated networks
• Number of loops of size h in growing scale-free networks
• Network Transitivity and Matrix Models
• Statistics of Cycles: How Loopy is your Network?
• The inhomogeneous evolution of subgraphs and cycles in complex networks

Cited by in corpus (2)

• Remarks on Bootstrap Percolation in Metric Networks
• Nonlinear Diffusion Through Large Complex Networks Containing Regular Subgraphs