This is a question about complex networks
We have various ways to measure the centrality or importance of a node.
$$\textrm{importance} :: \textrm{node} \rightarrow \mathbb{R}$$
The simplest such measure is the degree of the node.
There are some more advanced measures of centrality like eigenvector/katz/alpha centrality, pagerank, betweenness, etc…. These can find important nodes (in some sense) even when they have low degree.
We can also ask about the connection between two nodes
$$\textrm{connection} :: (\textrm{node}, \textrm{node}) \rightarrow \mathbb{R}$$
What measures do we have here? The analog to degree, the simplest importance measure, would be the weight of the edge between the two nodes (if any). Are there analogs of the other various centrality measures?
Question: What are known ways of computing the connection between two nodes?
Note – I am particularly interested in concepts which scale to very large graphs
Best Answer
Given a graph $G$, you can make a graph $H$ where the vertices of $H$ are the edges of $G$, and two vertices in $H$ are joined by an edge if and only if the corresponding edges in $G$ meet at a vertex of $G$. Then you can apply your favorite measure of centrality to vertices of $H$, and you'll actually be applying it to edges in $G$.
I don't know how good a measure of centrality you get this way, but I would think it would be worth a look.