Breaking the Small Cluster Barrier of Graph Clustering

TL;DR

This paper demonstrates that small clusters do not prevent the recovery of large clusters in graph clustering, introduces an iterative peeling algorithm, and extends results to partial observation settings.

Contribution

It refines the analysis of trace-norm based recovery to handle small clusters and proposes an iterative peeling strategy for almost complete cluster recovery.

Findings

Small clusters do not hinder large cluster recovery under mild conditions.

An iterative peeling algorithm effectively recovers almost all clusters.

The approach extends to partial observation scenarios with active learning.

Abstract

This paper investigates graph clustering in the planted cluster model in the presence of {\em small clusters}. Traditional results dictate that for an algorithm to provably correctly recover the clusters, {\em all} clusters must be sufficiently large (in particular, $\tilde{\Omega}(\sqrt{n})$ where $n$ is the number of nodes of the graph). We show that this is not really a restriction: by a more refined analysis of the trace-norm based recovery approach proposed in Jalali et al. (2011) and Chen et al. (2012), we prove that small clusters, under certain mild assumptions, do not hinder recovery of large ones. Based on this result, we further devise an iterative algorithm to recover {\em almost all clusters} via a "peeling strategy", i.e., recover large clusters first, leading to a reduced problem, and repeat this procedure. These results are extended to the {\em partial observation} setting, in which only a (chosen) part of the graph is observed.The peeling strategy gives rise to an active learning algorithm, in which edges adjacent to smaller clusters are queried more often as large clusters are learned (and removed). From a high level, this paper sheds novel insights on high-dimensional statistics and learning structured data, by presenting a structured matrix learning problem for which a one shot convex relaxation approach necessarily fails, but a carefully constructed sequence of convex relaxationsdoes the job.