Magnitude homology and Euler characteristics of directed acyclic graphs

What the math gives to ML

The paper provides a decategorified invariant for directed acyclic graphs derived from magnitude homology: alternating counts of ordered vertex chains at fixed directed path length, with cancellations governed by whether an intermediate vertex lies on a shortest path. This is a path-sensitive statistic that can distinguish DAGs with identical degree or layer-size profiles. A transferable neural-network use is to treat an MLP or sparse feed-forward network as a weighted DAG and regularize or monitor its length-resolved Euler signature, encouraging class-dependent computation paths rather than merely matching local connectivity. The main engineering risk is combinatorial growth and instability from continuous edge lengths, so the first implementation should use integer layer-hop costs, short-length truncation, and dynamic programming or sampled chains.