Hierarchy in a graph

Question

How can one compute or quantify the hierarchical nature of a given graph, if there is a hierarchy in that graph ? More specifically, i want to know if some sort of hierarchy existed in artificial neural network (with some number of hidden layers). and also want to measure that.

Answer 1

This is an interesting open ended question and so I'll answer loosely and off the cuff. So you want to find out if the graph is logically like a tree? There is no up or down in a graph, so probably what you are really looking for is a way to find the node or nodes in your graph that are the most highly connected to other highly connected nodes, and then take a certain perspective supposition and determine if a "tree" like graph makes sense using that node as the trunk or root of the tree. Another thing you could do is just choose any random node, suppose that is the root of the tree, and then see what happens. You can re-balance the tree if the node connections lead you to want to do that based on a certain number of connections or traversals, and you can attempt to find the "real root" - if there is such a thing. If you re-balance a certain number of times, or detect a circular path has been traversed, you may decide that the graph is not very hierarchical at all. If you do find a "real root", then you might then decide to look for depth, avg. branch numbers, balance stats, etc. If you refine the question, I'll refine my answer.