# 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.