Geometric approach to graph magnitude homology

Asao, Yasuhiko; Izumihara, Kengo

doi:10.4310/hha.2021.v23.n1.a16

Cited by 8 publications

(18 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It should be noted that this space has already been dealt with in previous works by Hepworth-Willerton [14] (for graphs) and Bottinelli-Kaiser [5] (for metric spaces). Asao-Izumihara [3] has also constructed a closely related space for graphs. One of the most crucial points in [3] was the introduction of the "time parameter t" to specify the vertex (x t , t) in the simplicial complex.…”

Section: Introductionmentioning

confidence: 99%

“…Asao-Izumihara [3] has also constructed a closely related space for graphs. One of the most crucial points in [3] was the introduction of the "time parameter t" to specify the vertex (x t , t) in the simplicial complex.…”

Section: Introductionmentioning

confidence: 99%

“…• The magnitude homotopy type is the double suspension of the space constructed by Asao-Izumihara [3] (Theorem 4.21).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Causal order complex and magnitude homotopy type of metric spaces

Yu¹,

Yoshinaga²

2023

Preprint

View full text Add to dashboard Cite

In this paper, we construct a pointed CW complex called the magnitude homotopy type for a given metric space X and a real parameter ℓ ≥ 0. This space is roughly consisting of all paths of length ℓ and has the reduced homology group that is isomorphic to the magnitude homology group of X.To construct the magnitude homotopy type, we consider the poset structure on the spacetime X × R defined by causal (time-or light-like) relations. The magnitude homotopy type is defined as the quotient of the order complex of an intervals on X × R by a certain subcomplex.The magnitude homotopy type gives a covariant functor from the category of metric spaces with 1-Lipschitz maps to the category of pointed topological spaces. The magnitude homotopy type also has a "path integral" like expression for certain metric spaces.By applying discrete Morse theory to the magnitude homotopy type, we obtain a new proof of the Mayer-Vietoris type theorem and several new results including the invariance of the magnitude under sycamore twist of finite metric spaces. Contents

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Causal order complex and magnitude homotopy type of metric spaces

Yu¹,

Yoshinaga²

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…However, whole picture of the behavior of magnitude is unrevealed, and that is attracting people in several areas of mathematics. In particular, magnitude of finite graphs, which takes values in formal power series with Z-coefficients, is studied by several authors so far ( [1], [3], [8], [9], [11]). Throughout this article, we call a finite, simple, and undirected graph without loops just a graph.…”

Section: Introductionmentioning

confidence: 99%

Girth, magnitude homology, and phase transition of diagonality

Asao¹,

Hiraoka²,

Kanazawa³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…This paper is organized as follows. In §2, we recall the definition of magnitude and magnitude homology of graphs, and introduce Asao-Izumihara complex following [2]. In §3, We describe the homotopy type of the Asao-Izumihara complex corresponding to pawful graphs.…”

Section: Introductionmentioning

confidence: 99%

Magnitude homology of graphs and discrete Morse theory on Asao-Izumihara complexes

Tajima¹,

Yoshinaga²

2021

Preprint

View full text Add to dashboard Cite

Recently, Asao and Izumihara introduced CW-complexes whose homology groups are isomorphic to direct summands of the graph magnitude homology group. In this paper, we study the homotopy type of the CWcomplexes in connection with the diagonality of magnitude homology groups. We prove that the Asao-Izumihara complex is homotopy equivalent to a wedge of spheres for pawful graphs introduced by Y. Gu. We also formulate a slight generalization of the notion of a pawful graph and find new non-pawful diagonal graphs of diameter 2.

show abstract

Geometric approach to graph magnitude homology

Cited by 8 publications

References 20 publications

Causal order complex and magnitude homotopy type of metric spaces

Causal order complex and magnitude homotopy type of metric spaces

Girth, magnitude homology, and phase transition of diagonality

Magnitude homology of graphs and discrete Morse theory on Asao-Izumihara complexes

Contact Info

Product

Resources

About