Centers for Random Walks on Trees

We consider two distinct centers which arise in measuring how quickly a random walk on a tree mixes. Lovász and Winkler [8] 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.