Degree Relations of Triangles in Real-world Networks and Models

TL;DR

This paper investigates the structure of triangles in real-world networks, revealing distinct patterns of degree homogeneity in social and information networks, and highlighting the inadequacy of current models to replicate these features.

Contribution

It introduces the concept of degree-labeled triangles and compares their prevalence and structure across different network types and models.

Findings

Social networks have predominantly homogeneous triangles.

Information networks are dominated by heterogeneous triangles.

Most existing graph models fail to accurately reproduce observed triangle structures.

Abstract

Triangles are an important building block and distinguishing feature of real-world networks, but their structure is still poorly understood. Despite numerous reports on the abundance of triangles, there is very little information on what these triangles look like. We initiate the study of degree-labeled triangles -- specifically, degree homogeneity versus heterogeneity in triangles. This yields new insight into the structure of real-world graphs. We observe that networks coming from social and collaborative situations are dominated by homogeneous triangles, i.e., degrees of vertices in a triangle are quite similar to each other. On the other hand, information networks (e.g., web graphs) are dominated by heterogeneous triangles, i.e., the degrees in triangles are quite disparate. Surprisingly, nodes within the top 1% of degrees participate in the vast majority of triangles in heterogeneous graphs. We also ask the question of whether or not current graph models reproduce the types of triangles that are observed in real data and showed that most models fail to accurately capture these salient features.