Limitations in the spectral method for graph partitioning: detectability threshold and localization of eigenvectors

TL;DR

This paper analyzes the spectral method's effectiveness in graph partitioning, identifying a detectability threshold and examining eigenvector localization effects, revealing a significant gap from Bayesian inference in sparse graphs.

Contribution

It estimates the detectability threshold for spectral methods in sparse graphs and investigates the impact of eigenvector localization on partitioning performance.

Findings

Detectability threshold estimated for spectral methods in sparse graphs.

Significant gap between spectral and Bayesian thresholds in sparse graphs.

Gap diminishes as graphs become denser.

Abstract

Investigating the performance of different methods is a fundamental problem in graph partitioning. In this paper, we estimate the so-called detectability threshold for the spectral method with both unnormalized and normalized Laplacians in sparse graphs. The detectability threshold is the critical point at which the result of the spectral method is completely uncorrelated to the planted partition. We also analyze whether the localization of eigenvectors affects the partitioning performance in the detectable region. We use the replica method, which is often used in the field of spin-glass theory, and focus on the case of bisection. We show that the gap between the estimated threshold for the spectral method and the threshold obtained from Bayesian inference is considerable in sparse graphs, even without eigenvector localization. This gap closes in a dense limit.