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Morse Theory and an impossibility theorem for graph clustering

Morse Theory and an impossibility theorem for graph clustering
Morse Theory and an impossibility theorem for graph clustering
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.
Sanchez Garcia, Ruben
8246cea2-ae1c-44f2-94e9-bacc9371c3ed
Strazzeri, Fabio
2fa6d25b-1ab5-43b9-a21c-c1e1454d0cb1
Sanchez Garcia, Ruben
8246cea2-ae1c-44f2-94e9-bacc9371c3ed
Strazzeri, Fabio
2fa6d25b-1ab5-43b9-a21c-c1e1454d0cb1

Sanchez Garcia, Ruben and Strazzeri, Fabio (2018) Morse Theory and an impossibility theorem for graph clustering. arXiv. (In Press)

Record type: Article

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.

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More information

Submitted date: 16 June 2018
Accepted/In Press date: 16 June 2018

Identifiers

Local EPrints ID: 430616
URI: http://eprints.soton.ac.uk/id/eprint/430616
PURE UUID: 4f01e96e-f753-4f64-8a9c-585103610764
ORCID for Ruben Sanchez Garcia: ORCID iD orcid.org/0000-0001-6479-3028

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Date deposited: 07 May 2019 16:30
Last modified: 22 Nov 2021 02:59

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Contributors

Author: Fabio Strazzeri

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