In a simple, unweighted network, I immediately think of the traditional density metric: ratio of realized edges over possible edges. We can also entertain more nuanced values, like number of nodes in cliques over the total number of nodes. Even median path length is a decent measure of density. When we have edge weights that represent "length" or some such, possibilities expand.
These graph-theoretic networks can exist happily without worrying about spatial embedding. (Unless you want to get into that in a very precise way.) The whole point of graph isomorphism is that we want to think of graphs as isomorphic regardless of how they are drawn.
I now spend some time each week at Civic Projects, a planning consulting firm. We're working on a project that is all about overlapping networks in urban space. How does the network of bike paths commingle with the network of car-centric streets? How do networks of wayfinding systems intertwine with networks of lighting infrastructure?
These networks are much different from ones I was used to in college. They are inherently spatially embedded and human-centric. Isomorphism doesn't matter: these are networks with anecdotal, instantiated importance.
Urban infrastructure networks also range widely in type. If we want to talk about the "network" of parking infrastructure, raw count of parking spots per unit area is a decently descriptive metric. Node-based metrics aren't very good at representing the density of, say, a bicycle path network, because these metrics can collapse travel distances. Instead, we want something that captures the travel times and interconnectedness. In lighting, if we treat each source of light as a node in the network, some measure of bulk distance between elements helps grasp and what needs to be measured for best human use.
I'm coming around to three major types of urban networks. (Nothing novel here, just useful for classification.)
Each of these necessitates a different strategy for density.
Context-dependent answers are necessary to make effective, well-reasoned, and responsible metrics for these types. And sometimes, a particular street feature doesn't fit neatly into one category. Trash cans, for instance—we want to consider both an areal density, but also think carefully about the causes for litter—maybe line-of-sight to a receptacle—which falls more into a path density representation. We might need to explore metrics of both forms to make an intelligent decision regarding a proposed system.
Why is any of this useful?
Lots of planners want rules-of-thumb when pitching concepts and making cost estimations. Infrastructure network statistics can also help codify best practices for urban design.
Lots more research required in this area.
(colophon: I am now at SIAM CSE 2017 to run a workshop on social systems modeling, through the Broader Engagement program. Brain getting back into math mode.)