I am an undergraduate studying economics and mathematics. I've never studied graph theory formally (only briefly in my spare time) and as such I don't have formal terms for what I'm clumsily trying to describe below. That's where I need help.
Suppose we have a graph $G$ with a set of verticies $R=\{ R_1,R_2,...,R_{n} \}$, and a set of weighted edges $E=\{ E_1,E_2,...,E_m \}$, where edge $E_i$ has weight $e_i$. Now, suppose further that each vertex $R_j$ has an associated value $r_j$.
I would like to assign a number $M_i$ to each vertex $R_i$ that captures its "access" to every other vertex's value $r_j$. The farther away vertex $R_{i}$ is from vertex $R_j$, the less impact $r_j$ has on $M_i$. I've made a simple example to try and show what I mean by that. The vertex $R_j$ and its associated value $r_j$ are represented by $R_j(r_j)$.
Vertex $R_3$ would have the value $M_3=\frac{1}{5} \cdot 100 + \frac{1}{2} \cdot 20$, vertex $R_2$ would have the value $M_2=\frac{1}{2} \cdot 50 + \frac{1}{2+5} \cdot 100$, and vertex $R_1$ would have the value $M_1 = \frac{1}{5} \cdot 50 + \frac{1}{5+2} \cdot 20$. What I'm trying to capture of course is a cost for distance.
Things obviously become more complicated the more verticies there are. I would want to find the shortest path from $R_{\alpha}$ to $R_{\beta}$ before including $r_{\beta}$ in the measure $M_{\alpha}$. The process for determining $M_i$ would be
- Find the shortest path from $R_i$ to $R_j$ for all $i \neq j$.
- Let $d_{ij}$ be the sum of the weights of the edges that make up the shortest path between $R_i$ and $R_j$.
- Then $M_i = \sum_{j=1\neq i}^{n} \frac{1}{d_{ij}} \cdot r_j$
What is this sort of process called? I imagine that it provides a measure of how "connected" a given node is in a network that is comprised of nodes that are not all valued equally nor of equal distance apart.
My context (roughly): Attempting to model the change in market access of regional economies in 19th century United States as they became more interconnected by railways. A regional economy would gain greater access to other regional economies as more railways connected them, but distance would increase costs.
You are trying to measure something like the conductance (or, inversely, the resistance) between the two node). The canonical (elementary) reference on this is Doyle and Snell's random walks and electrical network, though Bollobas' "Modern graph theory" has a good discussion as well.