Shortest paths through a reinforced random walk

In this article, we first study random walks on special electric networks with unit resistors and use four methods to solve Dirichlet problems in two dimensions. Then we study random walks on general resistor networks and give probabilistic interpretations to the qualities in electric network. In order to understand the movements of ants in an electric network, we describe a random walk model where ants moving through the network do not modify their path. After making a modification to this model by involving conductivity, we build current reinforced random walk model since ants are made to respond the current they generate. Here we allow ants to iteratively improve the path they take so that they can find the shortest path. Two algorithms based on exponential distribution and Poisson distribution are built and compared. We will look at the results about shortest path finding by random walks and implement simulation of two algorithms for shortest path problems.