Geometric approach to graph magnitude homology

Asao, Yasuhiko; Izumihara, Kengo

doi:10.48550/arxiv.2003.08058

Cited by 2 publications

(1 citation statement)

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“…(4) What can be said about the geometric meaning of H n,ℓ (X) for n > 2? Progress has been made since the appearance of the preprint version of this paper; see the works cited at the start of section 4, as well as [AI20] for the case of graphs. But much remains to be understood.…”

Section: Open Problemsmentioning

confidence: 99%

Magnitude homology of enriched categories and metric spaces

Leinster

Shulman

2021

Algebr. Geom. Topol.

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Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as [0, ∞)-enriched categories. We show that in many cases magnitude can be categorified to a homology theory for enriched categories, which we call magnitude homology (in fact, it is a special sort of Hochschild homology), whose graded Euler characteristic is the magnitude. Magnitude homology of metric spaces generalizes the Hepworth-Willerton magnitude homology of graphs, and detects geometric information such as convexity.

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Section: Open Problemsmentioning

confidence: 99%

Magnitude homology of enriched categories and metric spaces

Leinster

Shulman

2021

Algebr. Geom. Topol.

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Girth, magnitude homology, and phase transition of diagonality

Asao¹,

Hiraoka²,

Kanazawa³

2021

Preprint

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This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. Magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, respetively, 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 behavior 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 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.

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Geometric approach to graph magnitude homology

Cited by 2 publications

References 5 publications

Magnitude homology of enriched categories and metric spaces

Magnitude homology of enriched categories and metric spaces

Girth, magnitude homology, and phase transition of diagonality

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