Decompositions of Triangle-Dense Graphs

TL;DR

This paper explores the structure of triangle-dense graphs, showing they can be decomposed into dense subgraphs, and presents an algorithm that effectively recovers clusterings in certain stable instances.

Contribution

It provides a constructive decomposition of triangle-dense graphs into dense radius-2 subgraphs and demonstrates an algorithm for clustering in stable k-median problems.

Findings

Significant portions of triangle-dense graphs are unions of dense, radius-2 subgraphs.

The decomposition quantifies how these graphs resemble unions of cliques.

The algorithm successfully recovers planted clusterings in approximation-stable k-median instances.

Abstract

High triangle density -- the graph property stating that a constant fraction of two-hop paths belong to a triangle -- is a common signature of social networks. This paper studies triangle-dense graphs from a structural perspective. We prove constructively that significant portions of a triangle-dense graph are contained in a disjoint union of dense, radius 2 subgraphs. This result quantifies the extent to which triangle-dense graphs resemble unions of cliques. We also show that our algorithm recovers planted clusterings in approximation-stable k-median instances.