Application of random walks on graphs

Let $ G$ be an electrical network, i.e. a graph with nodes $ \{1,2,\dots,n\}$ in which to every edge $ E_{i,j}$ a conductivity $ \sigma_{i,j}$ is assigned. Suppose an electrical current $ I$ enters the network at node 1 and exists the network at node n (the current may enter and exit from multiple nodes as well). If $ \sigma_{i,j}$ is known, then it is an elementary problem to compute the induced current $ I$ along all edges of the graph $ G$ . Here $ I=(I_{i,j})$ is an $ n \times n$ matrix where $ I_{i,j}$ is the current flowing from node $ i$ to node $ j$ . Note that $ I_{i,j}=-I_{j,i}$ .

Suppose that only the matrix $ |I|=(|I_{i,j}|)$ is known. Assume also that the current entering the network at node 1 and exiting the network at node n is also known. We have developed an algorithm which determines the matrix $ I$ from the knowledge of $ |I|$ , as well as the class of conductivities $ \sigma=(\sigma_{i,j})$ which could produce the current $ I$ over the network. Indeed with this method one can design electrical networks (determine $ \sigma=(\sigma_{i,j})$ ) with prescribed $ |J|=|J_{i,j}|$ , or determine transition probabilities $ P=(p_{i,j})$ in random walk models, where $ p_{i,j}$ is the probability that a random walker takes an step from $ i$ to $ j$ , with prescribed net number of times the walker passes along each edge of the graph, i.e. $ |W_{i,j}-W_{j,i}|$ . Here $ W_{i,j}$ is the expected number of times the walker walks from node $ i$ to $ j$ .

Since electrical networks and random walks on graphs, and other diffusion problems on networks have numerous applications, I wonder if this method has any interesting application in problems that could be modeled by random walks on graphs or electrical networks.