Centers for Random Walks on Trees

Abstract

We consider two distinct centers which arise in measuring how quickly a random walk on a tree mixes. Lovasz and Winkler [Efficient stopping rules for Markov chains, in Proceedings of the 27th ACM Symposium on the Theory of Computing, 1995, pp. 76-82] point out that stopping rules which “look where they are going” (rather than simply walking a fixed number of steps) can achieve a desired distribution exactly and efficiently. Considering an optimal stopping rule that reflects some aspect of mixing, we can use the expected length of this rule as a mixing measure. On trees, a number of these mixing measures identify particular nodes with central properties. In this context, we study a variety of natural notions of centrality. Each of these criteria identifies the barycenter of the tree as the “average” center and the newly defined focus as the “extremal” center.