Eulerian magnitude homology: subgraph structure and random graphs

In this paper we explore the connection between the ranks of the magnitude homology groups of a graph and the structure of its subgraphs. To this end, we introduce variants of magnitude homology called eulerian magnitude homology and discriminant magnitude homology. Leveraging the combinatorics of the differential in magnitude homology, we illustrate a close relationship between the ranks of the eulerian magnitude homology groups on the first diagonal and counts of subgraphs which fall in specific classes. We leverage these tools to study limiting behavior of the eulerian magnitude homology groups for Erdos-Renyi random graphs and random geometric graphs, producing for both models a vanishing threshold for the eulerian magnitude homology groups on the first diagonal. This in turn provides a characterization of the generators for the corresponding magnitude homology groups. Finally, we develop an explicit asymptotic estimate the expected rank of eulerian magnitude homology along the first diagonal for these random graph models.