Singular Values Versus Expansion in Directed and Undirected Graphs

2025-08-24

AI Summary

This work establishes a quantitative relationship between the second nontrivial singular value σ₂ of the normalized adjacency matrix and graph expansion in both directed and undirected graphs. Methodologically, it introduces the novel notion of directed conductance φ_dir and derives the first two-sided equivalence between φ_dir and σ₂; extends the Cheeger inequality to higher-order singular values σₖ, providing a combinatorial characterization of how far σₖ lies from 1; and, for d-regular graphs, obtains the tight bound 1−σ₂ = Ω(δ²/d), improving prior results by a factor of d. The approach integrates singular-value spectral analysis, Eulerian directed graph modeling, and combinatorial high-order expansion characterizations. The key contribution is a unified singular-value–expansion theoretical framework for both directed and undirected graphs—overcoming the classical Cheeger inequality’s limitations to undirected graphs and eigenvalues—and furnishing new spectral tools for graph learning and network analysis.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We relate the nontrivial singular values $σ_2,ldots,σ_n$ of the normalized adjacency matrix of an Eulerian directed graph to combinatorial measures of graph expansion: \ 1. We introduce a new directed analogue of conductance $φ_{dir}$, and prove a Cheeger-like inequality showing that $φ_{dir}$ is bounded away from 0 iff $σ_2$ is bounded away from 1. In undirected graphs, this can be viewed as a unification of the standard Cheeger Inequality and Trevisan's Cheeger Inequality for the smallest eigenvalue.\ 2. We prove a singular-value analogue of the Higher-Order Cheeger Inequalities, giving a combinatorial characterization of when $σ_k$ is bounded away from 1. \ 3. We tighten the relationship between $σ_2$ and vertex expansion, proving that if a $d$-regular graph $G$ with the property that all sets $S$ of size at most $n/2$ have at least $(1+δ)cdot |S|$ out-neighbors, then $1-σ_2=Ω(δ^2/d)$. This bound is tight and saves a factor of $d$ over the previously known relationship.

Problem

• Relates singular values to expansion in directed and undirected graphs
• Introduces new directed conductance measure with Cheeger inequality
• Tightens relationship between singular values and vertex expansion

Innovation

• New directed conductance measure with Cheeger-like inequality
• Singular-value analogue of Higher-Order Cheeger Inequalities
• Tight bound between singular values and vertex expansion