Random Walks on Colored Graphs

We initiate a study of random walks on undirected graphs with colored edges. In our model, a sequence of colors is specified before the walk begins, and it dictates the color of edge to be followed at each step. We give tight upper and lower bounds on the expected cover time of a random walk on an undirected graph with colored edges. We show that, in general, graphs with two colors have exponential expected cover time, and graphs with three or more colors have doubly-exponential expected cover time. We also give polynomial bounds on the expected cover time in a number of interesting special cases. We describe applications of our results to understanding the dominant eigenvectors of products and weighted averages of stochastic matrices, and to problems on time-inhomogeneous Markov chains. 1 Introduction A colored graph is a set of n nodes with k distinctly-colored sets of undirected edges. Let the colors be represented by the numbers 1; 2; : : : ; k. An infinite sequence C = C 1 C 2 C 3..