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Decentralized search on spheres using small-world Markov chains: expected hitting times and structural properties

Part of: Markov processes Mathematical sociology Algorithms - Computer Science

Published online by Cambridge University Press:  01 July 2016

Archis Ghate
Archis Ghate*
Affiliation:
University of Washington
*
∗ Postal address: Industrial Engineering, University of Washington, Box 352650, Seattle, WA 98195-2650, USA. Email address: archis@u.washington.edu
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Abstract

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We build a family of Markov chains on a sphere using distance-based long-range connection probabilities to model the decentralized message-passing problem that has recently gained significant attention in the small-world literature. Starting at an arbitrary source point on the sphere, the expected message delivery time to an arbitrary target on the sphere is characterized by a particular expected hitting time of our Markov chains. We prove that, within this family, there is a unique efficient Markov chain whose expected hitting time is polylogarithmic in the relative size of the sphere. For all other chains, this expected hitting time is at least polynomial. We conclude by defining two structural properties, called scale invariance and steady improvement, of the probability density function of long-range connections and prove that they are sufficient and necessary for efficient decentralized message delivery.

Keywords

Small-world problemMarkov chain

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 68W40: Analysis of algorithms 91D30: Social networks
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 40 , Issue 4 , December 2008 , pp. 966 - 978
Copyright
Copyright © Applied Probability Trust 2008 

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