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

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 Erdős-Rényi graphs, and our results are on obtaining relevant upper bounds. These are also complemented with observations based on Monte Carlo simulations. Besides the Erdős-Rényi case, we also do simulation-based analysis for random regular graphs, the Barabási-Albert model and the Watts-Strogatz model.

💡 Research Summary

The manuscript introduces two global graph descriptors – the degree index (DI) and the clustering index (CI) – and investigates their behavior in several random‑graph ensembles, with a primary focus on Erdős–Rényi (ER) graphs.

Definitions.
For a simple graph (G=(V,E)) with (n=|V|), let (d_i) be the degree of vertex (i) and (C(i)) its local clustering coefficient (the fraction of closed triples among all possible triples formed by its neighbors). The authors define, for (\alpha\in{1,2}):
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