  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.
