I'm going through Graph Theory by Bondy, U. S. R. Murty and trying to get a grasp of basic graph theory, and am looking for literature (if it exists) that deals with graphs in which there is a " function attached to each node/edge" of sort. An example would be a situation (or if you could call it a game) where we start from a node, and where you can move between nodes in the fashion as in directed graphs, but in addition to that, moving once increases a time counter, and the said function is a function of that counter. The nodes or the edges could change in weight, as well as the direction of the edge, or a node could become inaccessible altogether depending on the input of the function.
More specifically, I'm looking to model a situation where we apply the said function to a (possibly) unique ordered set attached with each node/edge, and where the function at, say $T=n$, gives $n^{th}$ element in the set.
I know what I'm asking for could be way too specific, but any help or even a nudge in right direction would be appreciated.