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. 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.

Magnitude homology and Path homology

“…Magnitude homology of a graph MH ℓ k (g) is first introduced by Hepworth-Willerton in [4] as a categorification of magnitude, a categorical analogue of Euler characteristic introduced by Leinster ( [6]). Several authors show interesting properties of magnitude homology, which are similar to those of singular homology theory ( [1], [4], [5], [9]). However, only a few relation to the other graph invariants have been shown so far.…”

“…In this subsection, we show some properties of reduced path homology by using following fact proved in [1]. Note that the tuple (x 0 , x 1 , x 0 , .…”

“…However, only a few relation to the other graph invariants have been shown so far. For example, the girth of a graph controls magnitude homology to some extent ( [1]).…”

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

“…The diagonality of a graph is an important feature, because the rank of magnitude homology is determined by only the magnitude of G if G is diagonal. There are several recent studies on the diagonality [1,3,5,6,11]. Among other, let us recall two results concerning diagonality.…”

“…Among other, let us recall two results concerning diagonality. The first one is a result by Asao, Hiraoka and Kanazawa [1,Theorem 1.5], which asserts that if G is diagonal and is not a tree, then every edge is contained in a cycle of length ≤ 4. This gives a necessary condition for a graph to be diagonal.…”