I'm working with road networks, in which each edge is a physical street segment with a length attribute. Nodes represent junctions. However, my question should be generalizable to any weighted graph.
I'm interested in a specific measure of 'branchiness' that I have devised to normalise a density estimate. I would like to know if this corresponds to, or is related to, any pre-existing measures. I'm no expert in network science, and as such don't know what search terms to use.
The measure is best described with a diagram:
Start with a point, $a$, somewhere on the graph. Move an increasing distance away from $a$ and measure the number of shortest-distance branches that are possible at this distance. Once a branch terminates (as is the case for cul de sacs/dead ends in the road network), it is no longer counted. Repeat for many points on the network (e.g. point $b$ illustrated in the figure).
If I generate many traces for many points on the network, as shown in the top panel of the figure, the mean trace should appear smooth(?). I'm interested in whether this mean "distance<->branches" relationship has a name. It seems to be related to betweenness, but I don't think it's the same thing.