Statistical Mechanics of Semi-Supervised Clustering in Sparse Graphs

TL;DR

This paper provides a theoretical analysis of semi-supervised clustering in sparse graphs, showing how pairwise constraints influence detection thresholds and cluster recoverability.

Contribution

It offers a novel theoretical framework for understanding the effects of pairwise constraints on clustering in sparse graphs, extending previous unsupervised results.

Findings

Constraints shift detection thresholds at low densities.

High-density constraints suppress criticality and detection thresholds.

Adding constraints does not automatically improve clustering accuracy.

Abstract

We theoretically study semi-supervised clustering in sparse graphs in the presence of pairwise constraints on the cluster assignments of nodes. We focus on bi-cluster graphs, and study the impact of semi-supervision for varying constraint density and overlap between the clusters. Recent results for unsupervised clustering in sparse graphs indicate that there is a critical ratio of within-cluster and between-cluster connectivities below which clusters cannot be recovered with better than random accuracy. The goal of this paper is to examine the impact of pairwise constraints on the clustering accuracy. Our results suggests that the addition of constraints does not provide automatic improvement over the unsupervised case. When the density of the constraints is sufficiently small, their only impact is to shift the detection threshold while preserving the criticality. Conversely, if the density of (hard) constraints is above the percolation threshold, the criticality is suppressed and the detection threshold disappears.