On a 'Two Truths' Phenomenon in Spectral Graph Clustering

TL;DR

This paper investigates a 'Two Truths' phenomenon in spectral graph clustering, showing that Laplacian and adjacency spectral embeddings reveal different, meaningful community structures in brain connectome data.

Contribution

It uncovers and explains the 'Two Truths' phenomenon where different spectral embeddings highlight distinct underlying graph structures, supported by empirical MRI data analysis.

Findings

LSE captures hemisphere affinity structure

ASE reveals core-periphery structure

Different embeddings yield different meaningful clusters

Abstract

Clustering is concerned with coherently grouping observations without any explicit concept of true groupings. Spectral graph clustering - clustering the vertices of a graph based on their spectral embedding - is commonly approached via K-means (or, more generally, Gaussian mixture model) clustering composed with either Laplacian or Adjacency spectral embedding (LSE or ASE). Recent theoretical results provide new understanding of the problem and solutions, and lead us to a 'Two Truths' LSE vs. ASE spectral graph clustering phenomenon convincingly illustrated here via a diffusion MRI connectome data set: the different embedding methods yield different clustering results, with LSE capturing left hemisphere/right hemisphere affinity structure and ASE capturing gray matter/white matter core-periphery structure.