Fair and Biased Random Walks on Undirected Graphs and Related Entropies

Abstract

The entropy rates ofMarkov chains (randomwalks) defined on connected undirected graphs are well studied in many surveys. We study the entropy rates related to the first-passage time probability distributions of fair random walks, their relative (Kullback–Leibler) entropies, and the entropy related to two biased random walks –with the random absorption ofwalkers and the shortest paths randomwalks. We show that uncertainty of first-passage times quantified by the entropy rates characterizes the connectedness of the graph. The relative entropy derived for the biased random walks estimates the level of uncertainty between connectivity and connectedness – the local and global properties of nodes in the graph.