Can someone please point me in the direction of any theory on graphs where the edge weights are not scalars but represent some relation between the nodes that is a simple function of a single variable (simple, say piecewise linear).
In particular, I'm interested in various basic graph properties and also thinking of the graph as representing a network. So, for example, if the graph represented a communication network over time then the edge weights would be a function represented connectivity as a function of time, how do you find a valid path between nodes? I'm looking for help both on specific algorithms but also general theory if it exists?
I'm aware of time-extended networks where you explicitly expand out the dependence on the variable but, from what I've read, this incomplete and of limited applicability.