Random walks on colored graphs

The authors initiate a study of random walks on undirected graphs with colored edges. An adversary specifies a sequence of colors before the walk begins, and it dictates the color of edge to be followed at each step. They analyze the extent to which the adversary can influence the behavior of such a random walk, in terms of the expected cover time. They prove tight upper and lower bounds on the expected cover time of colored undirected graphs. They 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. They also give polynomial bounds on the expected cover time in a number of interesting special cases. They describe applications of these results to understanding the dominant eigenvectors of products and weighted averages of stochastic matrices, and to problems on time-inhomogeneous Markov chains