Magnitude and Topological Entropy of Digraphs

TL;DR

This paper introduces a novel notion of magnitude for flow graphs using topological entropy, linking it to volume entropy in digraphs and exploring applications in feature engineering.

Contribution

It develops a new framework connecting magnitude and topological entropy for digraphs, extending their use beyond metric spaces.

Findings

Magnitude for flow graphs based on topological entropy

Connection between magnitude and volume entropy in digraphs

Potential applications in feature engineering for graph data

Abstract

Magnitude and (co)weightings are quite general constructions in enriched categories, yet they have been developed almost exclusively in the context of Lawvere metric spaces. We construct a meaningful notion of magnitude for flow graphs based on the observation that topological entropy provides a suitable map into the max-plus semiring, and we outline its utility. Subsequently, we identify a separate point of contact between magnitude and topological entropy in digraphs that yields an analogue of volume entropy for geodesic flows. Finally, we sketch the utility of this construction for feature engineering in downstream applications with generic digraphs.