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Fair and Biased Random Walks on Undirected Graphs and Related Entropies

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Towards an Information Theory of Complex Networks

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.

MSC2000 Primary 82B41; Secondary 28D20, 68R10.

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Author information

Authors and Affiliations

  1. Bielefeld – Bonn Stochastic Research Center (BiBoS), University of Bielefeld, 100131, 33501, Bielefeld, Germany

    Philippe Blanchard

  2. The Center of Excellence Cognitive Interaction Technology (CITEC), University of Bielefeld, 100131, 33501, Bielefeld, Germany

    Dimitri Volchenkov

Authors
  1. Philippe Blanchard
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  2. Dimitri Volchenkov
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Correspondence to Dimitri Volchenkov .

Editor information

Editors and Affiliations

  1. Medizinische Informatik und Technik, Institute for Bioinformatics and Transla, UMIT-Private Universität für Gesundheits, Eduard Wallnöfer-Zentrum 1, Hall in Tirol, A-6060, Austria

    Matthias Dehmer

  2. Queen's University Belfast, School of Medicine, Dentistry, and Cell, Center for Cancer Research & Cell Biolog, Lisburn Road 97, Belfast, BT97BL, United Kingdom

    Frank Emmert-Streib

  3. Goethe-University Frankfurt am Main, Department of Philosophy and Historical, Center for Computing in the Humanities, Senckenberganlage 31, Frankfurt, 60325, Germany

    Alexander Mehler

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Blanchard, P., Volchenkov, D. (2011). Fair and Biased Random Walks on Undirected Graphs and Related Entropies. In: Dehmer, M., Emmert-Streib, F., Mehler, A. (eds) Towards an Information Theory of Complex Networks. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4904-3_13

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