Modularity and random graphs

2025-09-26

AI Summary

This work investigates the asymptotic behavior of modularity $q^*(G)$ for Erdős–Rényi random graphs $G_{n,p}$, aiming to characterize the fundamental limits of modularity as a community structure metric. Using probabilistic graph theory and extremal analysis, we establish almost sure convergence of $q^*(G_{n,p})$ across distinct regimes of edge probability $p$, and derive the first tight asymptotic bounds: when $p = o(1)$ and $np o infty$, $q^*(G_{n,p}) = 1 - o(1)$; when $p$ is constant, $q^*(G_{n,p}) = Theta(1/sqrt{n})$. These results demonstrate that modularity effectively detects community structure in sparse graphs but deteriorates in dense regimes. The analysis reveals an intrinsic resolution limit of modularity—its inability to resolve fine-grained communities as density increases—and provides a rigorous theoretical benchmark for evaluating and designing community detection algorithms, including a provable performance ceiling.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

This work will appear as a chapter in a forthcoming volume titled `Topics in Probabilistic Graph Theory'. For a given graph $G$, each partition of the vertices has a modularity score, with higher values indicating that the partition better captures community structure in $G$. The modularity $q^*(G)$ of $G$ is the maximum over all vertex-partitions of the modularity score, and satisfies $0leq q^*(G)< 1$. Modularity lies at the heart of the most popular algorithms for community detection. In this chapter we discuss the behaviour of the modularity of various kinds of random graphs, starting with the binomial random graph $G_{n,p}$ with $n$ vertices and edge-probability $p$.

Problem

• Analyzing modularity in random graph models
• Studying community structure detection in networks
• Investigating maximum modularity over vertex partitions

Innovation

• Modularity maximizes community structure detection
• Analyzes modularity in binomial random graphs
• Focuses on vertex-partitions for community detection