Let $g$ be a matrix corresponding to a directed graph so that $g_{ij}$ is the edge weight on an edge $ij$. How can we interpret $g^k$?

Question

I know that if $A$ is a 0-1 adjacency matrix then $[A^k]_{ij}$ is the number of walks of length $k$ from $i$ to $j$. Does this generalize nicely? The reason for this question is to interpret a result in Matthew O. Jackson's Social and Economic Networks where he considers a model in which each person $i$ assigns assign weights $g_{ij}$ to how much he cares about the action of person $j$ (actions are numbers in $[0,\infty))$. Each player chooses an action taking into account the actions of all the other people in the network with respect to the weights that they place on those actions. I'll leave the details, but he comes up with this

I am trying to understand what (9.9) is saying.