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Axioms for Distanceless Graph Partitioning

James Willson, Tandy Warnow

TL;DR

It is proved that clustering under the Constant Potts Model satisfies all the axioms, while Modularity clustering and iterative k-core both fail many axioms the authors pose.

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.

Axioms for Distanceless Graph Partitioning

TL;DR

It is proved that clustering under the Constant Potts Model satisfies all the axioms, while Modularity clustering and iterative k-core both fail many axioms the authors pose.

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.
Paper Structure (27 sections, 17 theorems, 24 equations, 1 figure, 1 table)

This paper contains 27 sections, 17 theorems, 24 equations, 1 figure, 1 table.

Table of Contents

  1. Abstract
  2. Author summary
  3. Introduction
  4. Background
  5. Our distanceless axioms
  6. Results
  7. Discussion
  8. Conclusion
  9. Appendix

Key Result

Lemma 1

If $M$ is a clustering method that satisfies Connectivity, then for some $n_0 \geq 1$, no clusters returned by $M$ of size at least $n_0$ have cut edges.

Figures (1)

  • Figure 1: Theoretical properties of IKC and IKC(no-mod). In each case, the network shown is one component of a network with two components, where the second component is a single edge (hence the shown component always has positive modularity). Green edges are added to a starting network, red edges are deleted from a starting network, and blue edges represent edges that are in the starting and final network. Subfigure (a) gives one component in a network $N_1$ where IKC and IKC(no-mod) both fail Standard Consistency. Subfigure (b) gives one component in a network $N_2$ where IKC and IKC(no-mod) both fail Refinement Consistency. Subfigure (c) gives an example of one component in a network $N_3$ where IKC will return only singleton clusters (due to its check for positive modularity). However, if the red edges were deleted, then IKC would return the 3-clique, establishing IKC fails Inter-Edge Consistency.

Theorems & Definitions (32)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 22 more