Girth, magnitude homology and phase transition of diagonality

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.

Magnitude homology and path homology

“…Magnitude homology can be defined for general metric spaces [13], and several authors show interesting geometric properties of that. It is remarkable that such properties are similar to those of singular homology theory ([1], [2], [3], [10], [11], [14], for example). However, only a few relations to the other invariants have been shown so far.…”

“…Remark The diagonality of other graphs is studied, for example, in [14]. It is known that the girth of any diagonal graph which is not a tree is 3 or 4 ([2] Corollary 1.6).…”

“…In this subsection, we show some properties of reduced path homology by using following fact proved in [2]. Note that the tuple $$(x_0, x_1, x_0, \dots )$$ obtained by $$(k+1)$$ times repeating distinct vertices $$x_0$$ and $$x_1$$ with $$\lbrace x_0, x_1\rbrace \in E(g)$$ is a homology cycle of $$\operatorname{MC}^{k}_k(\overrightarrow{g})$$ for any undirected graph $$g$$.…”

“…However, only a few relations to the other invariants have been shown so far. For example, the girth of a graph controls magnitude homology to some extent [2]. For geodesic metric spaces, magnitude homology indicates curvature boundedness or uniqueness of geodesics [1], [3].…”

“…We show that the inverse of Theorem 1.3 is not true in Example 8.10. We apply a result in [2] to show the following. See also Remark 8.16.…”