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

Ümit Işlak, Barış Yeşiloğlu

TL;DR

This work introduces two aggregate graph-heterogeneity measures, the alpha-level degree index $\mathrm{DI}_{\alpha}(\mathcal{G})$ and the novel alpha-level clustering index $\mathrm{CI}_{\alpha}(\mathcal{G})$, and analyzes their behavior in random graphs. It yields explicit or asymptotic expressions for the degree indices in Erdős–Rényi graphs, and establishes linear and constant upper bounds for the clustering indices, complemented by Monte Carlo investigations across random regular, Barabási–Albert, and Watts–Strogatz models. The results reveal distinct scaling: for Erdős–Rényi graphs, $\mathbb{E}[\mathrm{DI}_1]$ scales as $\Theta(n^{5/2})$ and $\mathbb{E}[\mathrm{DI}_2]$ as $\Theta(n^3)$, while $\mathbb{E}[\mathrm{CI}_1]$ grows linearly in $n$ and $\mathbb{E}[\mathrm{CI}_2]$ remains bounded. The study highlights the potential utility of these indices as features in graph-based classification and economic-crisis detection, with directions for exact asymptotics and real-network validation discussed for future work.

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 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.

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

TL;DR

This work introduces two aggregate graph-heterogeneity measures, the alpha-level degree index and the novel alpha-level clustering index , and analyzes their behavior in random graphs. It yields explicit or asymptotic expressions for the degree indices in Erdős–Rényi graphs, and establishes linear and constant upper bounds for the clustering indices, complemented by Monte Carlo investigations across random regular, Barabási–Albert, and Watts–Strogatz models. The results reveal distinct scaling: for Erdős–Rényi graphs, scales as and as , while grows linearly in and remains bounded. The study highlights the potential utility of these indices as features in graph-based classification and economic-crisis detection, with directions for exact asymptotics and real-network validation discussed for future work.

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 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.
Paper Structure (11 sections, 11 theorems, 77 equations, 9 figures)

This paper contains 11 sections, 11 theorems, 77 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Some basic observations on the clustering index
  3. Clustering index of Erdős-Rényi graphs
  4. Sublinearity of $\mathbb{E}[\mathrm{CI}_1(\mathcal{G})]$ and $\mathbb{E}[\mathrm{CI}_2(\mathcal{G})]$
  5. Further analysis of $\mathbb{E}[\mathrm{CI}_2(\mathcal{G})]$
  6. Degree index in Erdős-Rényi graphs
  7. Monte Carlo simulations for other random graph models
  8. Random graph models and implementation
  9. Results for the clustering index
  10. Results for the degree index
  11. Conclusion

Key Result

Lemma 2.1

Let $a_1, a_2, \ldots, a_n$ be real numbers on $[0,1]$. Then for any $\alpha \geq 1$, we have,

Figures (9)

  • Figure 1: A graph with $8$ vertices where $\mathrm{CI}_{\alpha} = 8^2/4 = 16$.
  • Figure 2: A graph with $d_i = 2$ for any $i$ (hence $\mathrm{DI}_{\alpha} = 0$) but $\mathrm{CI}_{\alpha}(\mathcal{G}) = n^2/4$.
  • Figure 3: A graph with $C(i) = 1$ for any $i$ (hence $\mathrm{CI}_{\alpha} (\mathcal{G})= 0$) but $\mathrm{DI}_{1}(\mathcal{G}) = \frac{n^2(n/2-3)}{4}$.
  • Figure 4: Average $\mathrm{CI}_1$ values of the Erdős-Rényi graph, Barabási-Albert graph, random regular graph and Watts-Strogatz graphs with edge densities 0.1 and 0.5.
  • Figure 5: Average $\mathrm{CI}_2$ values of the Erdős-Rényi graph, Barabási-Albert graph, random regular graph and Watts-Strogatz graphs with edge densities 0.1 and 0.5.
  • ...and 4 more figures

Theorems & Definitions (15)

  • Lemma 2.1
  • Corollary 2.1
  • Example 2.1
  • Lemma 2.2
  • Example 2.2
  • Example 2.3
  • Proposition 3.1
  • Theorem 3.1
  • Corollary 3.1
  • Proposition 3.2
  • ...and 5 more