Relations Between Adjacency and Modularity Graph Partitioning

TL;DR

This paper establishes a precise linear relationship between adjacency and modularity matrices, proposes an approximation method for eigenvectors, and demonstrates efficiency gains in clustering methods.

Contribution

It introduces a novel exact linear relationship and an approximation technique for eigenvectors, linking adjacency and modularity clustering.

Findings

Normalized adjacency clustering can be twice as efficient as modularity clustering

Derived the error bounds for the eigenvector approximation

Proved the equivalence between normalized adjacency and modularity clustering

Abstract

This paper develops the exact linear relationship between the leading eigenvector of the unnormalized modularity matrix and the eigenvectors of the adjacency matrix. We propose a method for approximating the leading eigenvector of the modularity matrix, and we derive the error of the approximation. There is also a complete proof of the equivalence between normalized adjacency clustering and normalized modularity clustering. Numerical experiments show that normalized adjacency clustering can be as twice efficient as normalized modularity clustering.