Suppose I have n points and a way to measure the pairwise (probably non-Euclidean) distance between them. I would like to have some way to measure the total "closeness" of my points, but I'm not really sure of the best way to do that.
First of all, I could see approaching this from some sort of analytical or topological angle (since I suppose I really do have a metric space) or from a graph theory perspective (I could consider each element a node of the complete weighted graph on n vertices such that the weight aligns with the distance between nodes). I'm leaning towards a graph theory approach but I can't articulate why.
One obvious approach that comes to mind is to simply take the sum of all pairwise distances. I don't know why, but that just seems too simple.
Another approach would be to find the minimal distance to traverse all nodes in the traveling salesperson sense. This might best align with an intuitive notion of closeness, but it obviously becomes intractable beyond a very small number of nodes (although in my case I happen to be dealing with a small number of nodes anyway).
A compromise on the TSP path length would perhaps be to find the weight of a minimal spanning tree. That should be performant and still somewhat intuitive, so it's probably the leading candidate right now.
Is there a standard approach to this thing that I am unaware of? Am I overlooking any better technique here?
As an aside, I'm curious whether or not the triangle inequality is necessary (since I'm not 100% sure that the distance metric I want really is a distance metric).
EDIT: I forgot to mention this, but my space need not have anything to do with $\mathbb R^{n}$, and indeed in the case I have in mind the space is finite.