Which metric measures the distance of two nodes in a graph, also accounting for the number of alternative routes I would like to measure the "closeness" of two nodes $i$ and $j$ on an undirected, unweighed graph. The ideal metric I'd be interested in has the flowing properties:


*

*Strictly increasing in the number of paths from $i$ to $j$

*Decreasing in the length of each path from $i$ to $j$


I was thinking of using the sum of the inverse of path-lengths from $i$ to $j$. This would have the desired properties. I am quite sure thought that this must already have a name. I failed at googleing it though.
Some background:
I'm trying to measure social proximity in a social network. The idea is that (i) the shorter the chain between two people and (ii) the more common friends there are, the more socially proximate they are.

Only a start an an answer, too long for a comment.
You're asking a well studied question. If you google

https://www.google.com/#q=measuring+proximity+in+graphs

you'll find ideas for lots of algorithms that address your problem.
I suggest you start here: http://www.cs.cmu.edu/~wcohen/10-802/fixed/Different_proximity_measurements_on_graphs.html
