Detectability of the spectral method for sparse graph partitioning

TL;DR

This paper reveals that the spectral method for graph partitioning has universal detectability across different resolution parameters, and identifies a phase transition leading to unpartitioned graphs at small resolution values.

Contribution

It unifies spectral method detectability analysis under a single framework and characterizes phase transitions related to the resolution parameter.

Findings

Spectral method detects communities regardless of resolution parameter.

A phase transition occurs at small resolution values, leading to no partition.

The framework unifies understanding of modularity maximization in graph partitioning.

Abstract

We show that modularity maximization with the resolution parameter offers a unifying framework of graph partitioning. In this framework, we demonstrate that the spectral method exhibits universal detectability, irrespective of the value of the resolution parameter, as long as the graph is partitioned. Furthermore, we show that when the resolution parameter is sufficiently small, a first-order phase transition occurs, resulting in the graph being unpartitioned.