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Girth, magnitude homology and phase transition of diagonality

Part of: Combinatorial probability Graph theory Homology and cohomology theories

Published online by Cambridge University Press:  09 February 2023

Yasuhiko Asao ,
Yasuaki Hiraoka Open the ORCID record for Yasuaki Hiraoka [Opens in a new window]  and
Shu Kanazawa Open the ORCID record for Shu Kanazawa [Opens in a new window]
Yasuhiko Asao
Affiliation:
Department of Applied Mathematics, Fukuoka University, Fukuoka, Japan (asao@fukuoka-u.ac.jp)
Yasuaki Hiraoka
Affiliation:
Kyoto University Institute for Advanced Study, WPI-ASHBi, Kyoto University, Kyoto, Japan Center for Advanced Intelligence Project, RIKEN, Saitama, Japan (hiraoka.yasuaki.6z@kyoto-u.ac.jp)
Shu Kanazawa
Affiliation:
Kyoto University Institute for Advanced Study, Kyoto University, Kyoto, Japan (kanazawa.shu.2w@kyoto-u.ac.jp)
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Abstract

This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. The magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, respectively, for finite metric spaces, first introduced by Leinster and Hepworth–Willerton. Several authors study them restricting to graphs with path metric, and some properties which are similar to the ordinary homology theory have come to light. However, the whole picture of their behaviour is still unrevealed, and it is expected that they catch some geometric properties of graphs. In this article, we show that the girth of graphs partially determines the magnitude homology, that is, the larger girth a graph has, the more homologies near the diagonal part vanish. Furthermore, applying this result to a typical random graph, we investigate how the diagonality of graphs varies statistically as the edge density increases. In particular, we show that there exists a phase transition phenomenon for the diagonality.

Keywords

Magnitude homologygirthrandom graphphase transition

MSC classification

Primary: 55N35: Other homology theories
Secondary: 05C31: Graph polynomials 05C80: Random graphs 60C05: Combinatorial probability
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 1 , February 2024 , pp. 221 - 247
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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