Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Cheeger Inequality for Size-Specific Conductance

Yufan Huang, David F. Gleich

TL;DR

This paper addresses the challenge of understanding the size-specific conductance profile of a graph by studying $\mu$-conductance $\phi_{\mu}$ and a corresponding spectral relaxation with optimum $\lambda_{\mu}$. It establishes a two-sided Cheeger-type inequality $2 \phi_{\mu} \ge \lambda_{\mu} \ge \frac{1}{4}\left( \frac{\mu^2 \phi_{\mu} + (\tfrac{1}{2}-\mu) \phi_0}{\mu^2 + \tfrac{1}{2}-\mu} \right)^2$, connecting the spectral program to size-constrained cuts and recovering classical Cheeger bounds in the appropriate limits. The paper also develops computational strategies, including an SDP relaxation and a Burer–Monteiro low-rank approach, to obtain scalable lower bounds and practical estimates for large graphs; a small computational example demonstrates the bounds' behavior as the size constraint varies. Overall, the work extends Cheeger-type relations to μ-conductance, enabling multi-scale graph clustering insights and tractable approximations for large networks.

Abstract

The $μ$-conductance measure proposed by Lovász and Simonovits is a size-specific conductance score that identifies the set with smallest conductance while disregarding those sets with volume smaller than a $μ$ fraction of the whole graph. Using $μ$-conductance enables us to study the network structures in new ways. In this manuscript we study a modified spectral cut for $μ$-conductance that is a natural relaxation of the integer program of $μ$-conductance and show that the optimum of this program has a two-sided Cheeger inequality with $μ$-conductance.

A Cheeger Inequality for Size-Specific Conductance

TL;DR

This paper addresses the challenge of understanding the size-specific conductance profile of a graph by studying -conductance and a corresponding spectral relaxation with optimum . It establishes a two-sided Cheeger-type inequality , connecting the spectral program to size-constrained cuts and recovering classical Cheeger bounds in the appropriate limits. The paper also develops computational strategies, including an SDP relaxation and a Burer–Monteiro low-rank approach, to obtain scalable lower bounds and practical estimates for large graphs; a small computational example demonstrates the bounds' behavior as the size constraint varies. Overall, the work extends Cheeger-type relations to μ-conductance, enabling multi-scale graph clustering insights and tractable approximations for large networks.

Abstract

The -conductance measure proposed by Lovász and Simonovits is a size-specific conductance score that identifies the set with smallest conductance while disregarding those sets with volume smaller than a fraction of the whole graph. Using -conductance enables us to study the network structures in new ways. In this manuscript we study a modified spectral cut for -conductance that is a natural relaxation of the integer program of -conductance and show that the optimum of this program has a two-sided Cheeger inequality with -conductance.
Paper Structure (10 sections, 4 theorems, 47 equations, 1 figure, 1 table)

This paper contains 10 sections, 4 theorems, 47 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cheeger Inequality for $\mu$-conductance
  4. The definition of $\mu$-conductance
  5. A spectral program for $\mu$-conductance
  6. Main Result
  7. Proof for the upper bound of $\lambda_\mu$
  8. Proof for lower bound of $\lambda_\mu$
  9. Computational considerations and examples
  10. Computational Example

Key Result

Theorem 3.1

Given a graph $G$ and a constant $0 \leq \mu \leq 1/2$, we have

Figures (1)

  • Figure 1: An example graph with 85 vertices and 568 edges (Left) and the bounds from \ref{['lem:cheeger-upper-bound']} computed on it for different $\mu$s (Right). To keep track of $\lambda_{\mu}$, the optimum of the spectral program, we compute its relaxation $\lambda_{\mu}^{\text{SDP}}$. The true $\lambda_{\mu}$ lies between the blue upper bound and the orange SDP relaxation. We see that the lower bound gets tighter in the large $\mu$ regime.

Theorems & Definitions (7)

  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof : Proof of \ref{['lem:core']}