Axioms for Distanceless Graph Partitioning

TL;DR

This paper introduces new axioms for unweighted graph partitioning, modifies Kleinberg's axioms, and evaluates clustering methods like the Constant Potts Model and Modularity against these axioms.

Contribution

It proposes axioms specific to unweighted graph partitioning and analyzes how existing methods satisfy or violate these axioms.

Findings

Constant Potts Model satisfies all proposed axioms.

Modularity clustering fails many axioms.

Iterative k-core also violates several axioms.

Abstract

In 2002, Kleinberg proposed three axioms for distance-based clustering, and proved that it was impossible for a clustering method to satisfy all three. While there has been much subsequent work examining and modifying these axioms for distance-based clustering, little work has been done to explore axioms relevant to the graph partitioning problem when the graph is unweighted and given without a distance matrix. Here, we propose and explore axioms for graph partitioning for this case, including modifications of Kleinberg's axioms and three others: two axioms relevant to the ``Resolution Limit'' and one addressing well-connectedness. We prove that clustering under the Constant Potts Model satisfies all the axioms, while Modularity clustering and iterative k-core both fail many axioms we pose. These theoretical properties of the clustering methods are relevant both for theoretical investigation as well as to practitioners considering which methods to use for their domain science studies.