Finding Dense Clusters via "Low Rank + Sparse" Decomposition

TL;DR

This paper proposes a convex optimization approach to detect dense clusters in graphs by decomposing adjacency matrices into low rank and sparse components, effectively identifying imperfect cliques.

Contribution

It introduces a novel method combining low rank and sparse decomposition to find dense clusters, extending recent convex optimization techniques for graph analysis.

Findings

Dense clusters can be identified if they are sufficiently large.

The method models imperfect cliques with missing edges as sparse components.

The approach is analyzed in a probabilistic setting for planted clusters.

Abstract

Finding "densely connected clusters" in a graph is in general an important and well studied problem in the literature \cite{Schaeffer}. It has various applications in pattern recognition, social networking and data mining \cite{Duda,Mishra}. Recently, Ames and Vavasis have suggested a novel method for finding cliques in a graph by using convex optimization over the adjacency matrix of the graph \cite{Ames, Ames2}. Also, there has been recent advances in decomposing a given matrix into its "low rank" and "sparse" components \cite{Candes, Chandra}. In this paper, inspired by these results, we view "densely connected clusters" as imperfect cliques, where imperfections correspond missing edges, which are relatively sparse. We analyze the problem in a probabilistic setting and aim to detect disjointly planted clusters. Our main result basically suggests that, one can find \emph{dense} clusters in a graph, as long as the clusters are sufficiently large. We conclude by discussing possible extensions and future research directions.