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