Possibility results for graph clustering: a novel consistency axiom
Possibility results for graph clustering: a novel consistency axiom
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.
Axiomatic clustering, Data clustering, Graph clustering, Morse flow, Morse theory
Sanchez Garcia, Ruben
8246cea2-ae1c-44f2-94e9-bacc9371c3ed
Strazzeri, Fabio
2fa6d25b-1ab5-43b9-a21c-c1e1454d0cb1
August 2022
Sanchez Garcia, Ruben
8246cea2-ae1c-44f2-94e9-bacc9371c3ed
Strazzeri, Fabio
2fa6d25b-1ab5-43b9-a21c-c1e1454d0cb1
Sanchez Garcia, Ruben and Strazzeri, Fabio
(2022)
Possibility results for graph clustering: a novel consistency axiom.
Pattern Recognition, 128, [108687].
(doi:10.1016/j.patcog.2022.108687).
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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Possibility_and_Impossibility_Theorems_for_Graph_Clustering__A_Novel_Consistency_Axiom_accepted
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More information
Accepted/In Press date: 2 April 2022
e-pub ahead of print date: 8 April 2022
Published date: August 2022
Additional Information:
Funding Information:
We thank Francisco Belchí Guillamón for valuable discussions about Monotonic Consistency and metrics. FS was supported by a PhD studentship by Mathematical Sciences and the Institute for Life Sciences at the University of Southampton. RSG was partially supported by The Alan Turing Institute under the EPSRC grant EP/N510129/.
Publisher Copyright:
© 2022
Keywords:
Axiomatic clustering, Data clustering, Graph clustering, Morse flow, Morse theory
Identifiers
Local EPrints ID: 456478
URI: http://eprints.soton.ac.uk/id/eprint/456478
ISSN: 0031-3203
PURE UUID: 10f45329-c2bc-4aac-82c9-080867820b68
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Date deposited: 03 May 2022 16:50
Last modified: 17 Mar 2024 03:21
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Author:
Fabio Strazzeri
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