Electric Networks and Commute Time

The equivalence of random walks on weighted graphs with reversible Markov chains has long been known. Another such correspondence exists between electric networks and these random walks. Results in the language of random walks have analogues in the language of electric networks and visa versa. We outline this correspondence and describe how to translate several of the major terms of electric networks into the language of random walks. Given a random walk on the weighted graph G = (V,E) and u, v ∈ V , the commute time Cuv between u and v is the expected number of steps for a random walk starting at u to pass through v and return to u. The cover time CG is the expected number of steps for a random walk to hit every point. In the second section, we prove that Cuv is equal to the product of 2|E| and the effective resistance between u and v in the corresponding electric network following an approach first demonstrated in [1]. The proof highlights several ideas previously discussed. We then use this result to prove bounds for CG. 1 Random Walks and Electric Networks The existence of a correspondence between random walks on graphs and electrical networks was first established in [4]. Both systems are governed by graphs which have values attached to the edges, weight and resistance respectively. More importantly, the physical process electric networks model is intimately related to walks on graphs. An electrical network governs the aggregate flow of electrons through the circuit. However, the behavior of an individual electron is thought to approximate a random walk through the nodes of the network. We can then view the electric network as Monte Carlo approximation of this random walk on a scale that eliminates all detectable error. With this in mind, we set out to discover how the aggregate behavior defines the underlying random walk. Our initial objective is to relate the weights on edges of the graph with the resistance of edges in the electric network. Let G = (V,E) be a connected weighted graph. Here, each