Possibility results for graph clustering: A novel consistency axiom

TL;DR

This paper introduces Monotonic Consistency, a new clustering property that avoids Kleinberg's impossibility theorem, and presents Morse Clustering, an algorithm inspired by Morse Theory that satisfies this property.

Contribution

The paper proposes Monotonic Consistency as a new axiom and introduces Morse Clustering, a novel algorithm that satisfies this axiom and extends Kleinberg's framework to sparse graphs.

Findings

Morse Clustering uncovers flow structures on graphs.

Morse Clustering satisfies Kleinberg's axioms with Monotonic Consistency.

Impossibility results for Kleinberg's axioms are extended to sparse graphs.

Abstract

Kleinberg introduced three natural clustering properties, or axioms, and showed they cannot be simultaneously satisfied by any clustering algorithm. We present a new clustering property, Monotonic Consistency, which avoids the well-known problematic behaviour of Kleinberg's Consistency axiom, and the impossibility result. Namely, we describe a clustering algorithm, Morse Clustering, inspired by Morse Theory in Differential Topology, which satisfies Kleinberg's original axioms with Consistency replaced by Monotonic Consistency. Morse clustering uncovers the underlying flow structure on a set or graph and returns a partition into trees representing basins of attraction of critical vertices. We also generalise Kleinberg's axiomatic approach to sparse graphs, showing an impossibility result for Consistency, and a possibility result for Monotonic Consistency and Morse clustering.