Random graph model with power-law distributed triangle subgraphs

TL;DR

This paper analyzes the distribution and asymptotic behavior of triangles in a scale-free random graph model, revealing power-law distributions and phase transitions in clustering properties.

Contribution

It introduces a detailed analysis of triangle arrangements in a scale-free graph model, highlighting power-law distributions and phase transitions based on degree exponent.

Findings

Triangles follow a power-law distribution among vertices.

Clustering coefficient and number of triangles exhibit specific asymptotic behaviors.

Finite triangle density occurs at a critical degree exponent.

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 $\beta=2+1/3$.