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 <a href="http://www.maths.ed.ac.uk/~tl/">webpage</a>. Many of the entries with link to "discussions" refer to discussions taking place at the $n$-Category Caf&eacute; over a period of years.