I would say you could make good headway on this by looking over some of the research projects of Tom Leinster, Mark Meckes, and Simon Willerton (and others I may be forgetting), centering on various notions of size/magnitude for graphs, finite metric spaces, and various other structures. You could start by perusing the entries under "Size" on Leinster's webpage. Many of the entries with link to "discussions" refer to discussions taking place at the $n$-Category Café over a period of years.