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Fair and Biased Random Walks on Undirected Graphs and Related Entropies

Chapter

Abstract

The entropy rates of Markov chains (random walks) 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 of walkers and the shortest paths random walks. 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.

Keywords

Entropy of graphs First-passage times Random walks on graphs 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Bielefeld – Bonn Stochastic Research Center (BiBoS)University of BielefeldBielefeldGermany
  2. 2.The Center of Excellence Cognitive Interaction Technology (CITEC)University of BielefeldBielefeldGermany

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