Analysis of Clustering and Degree Index in Random Graphs and Complex Networks

TL;DR

This paper analyzes degree and clustering indices in various random graph models, providing theoretical bounds and simulation results to understand their properties in complex networks.

Contribution

It introduces a new clustering index, derives theoretical bounds for Erdős-Rényi graphs, and compares these indices across multiple network models.

Findings

Clustering index bounds are established for Erdős-Rényi graphs.

Monte Carlo simulations validate theoretical results.

Analysis extends to random regular, Barabási-Albert, and Watts-Strogatz models.

Abstract

The purpose of this paper is to analyze the degree index and clustering index in random graphs. The degree index in our setup is a certain measure of degree irregularity whose basic properties are well studied in the literature, and the corresponding theoretical analysis in a random graph setup turns out to be tractable. On the other hand, the clustering index, based on a similar reasoning, is first introduced in this manuscript. Computing exact expressions for the expected clustering index turns out to be more challenging even in the case of Erd\H{o}s-R\'enyi graphs, and our results are on obtaining relevant upper bounds. These are also complemented with observations based on Monte Carlo simulations. Besides the Erd\H{o}s-R\'enyi case, we also do simulation-based analysis for random regular graphs, the Barab\'asi-Albert model and the Watts-Strogatz model.