Categorifying the magnitude of a graph

Abstract

The magnitude of a graph can be thought of as an integer power series associated to a graph; Leinster introduced it using his idea of magnitude of a metric space. Here we introduce a bigraded homology theory for graphs which has the magnitude as its graded Euler characteristic. This is a categorification of the magnitude in the same spirit as Khovanov homology is a categorification of the Jones polynomial. We show how properties of magnitude proved by Leinster categorify to properties such as a Künneth Theorem and a Mayer-Vietoris Theorem. We prove that joins of graphs have their homology supported on the diagonal. Finally, we give various computer calculated examples.2010 Mathematics Subject Classification. 55N35, 05C31.

Smoothness filtration of the magnitude complex

“…As preliminary, we introduce some basic notions about magnitude complexes. For more detail, we refer to [4,6,9], although we are using some different notations.…”

“…Then the magnitude [8,9] of the metric space is defined to be the sum Mag(X, d) = x,y∈X M (x, y) of the entries in the matrix Z −1 = M = (M (x, y)) inverse to Z. The crucial fact [4,9] is the formula justifying that "the magnitude homology is the categorification of the magnitude":…”

“…the magnitude cohomology. This is anticipated in [4], and is investigated in [3] together with its non-commutative product. Dualizing the filtration of the magnitude chain complex, we get a filtration of the magnitude cochain complex.…”

“…The magnitude homology is a homology group of a chain complex constructed from any metric space. As a categorification of the magnitude of a metric space [8], this notion is first introduced to finite metric spaces associated to graphs with the shortest path length by Hepworth and Willerton [4], and then generalized to general metric spaces by Leinster and Shulman [9]. Since the introduction, a number of features of the magnitude homology have been uncovered recently [3,5,6,11].…”

On the Magnitude and Intrinsic Volumes of a Convex Body in Euclidean Space

“…See [18] for a survey of connections between magnitude and geometry. In other directions, magnitude has connections to graph invariants [16], theoretical ecology [17,26], and homology theory [3,11,12,19,24].…”