Graph centrality measures are a pretty interesting and ongoing topic of research in mathematics, yes. I'm familiar with methods based on eigenspectra of the matrix exponential (and some relevance to Krylov subspace methods), but that's really it.
But I guess there will be some arbitrariness involved anyway, even when mathematicians look at the problem. E.g. why would one measure be better than another?
Certainly not to record the relations or visualize them, but social network analysis certainly falls into the realm of mathematics.
In this case the mathematician decided on his own to make the diagram, it isn't an answer to some external need.