Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T20:12:44.445Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Markov processes Algorithms - Computer Science Mathematical sociology

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Albert, R. and Barabasi, A. L. (2002). Statistical mechanics of complex networks. Rev. Modern Phys. 74, 47–97.CrossRefGoogle Scholar
Barabasi, A. L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509–512.Google Scholar
Franceschetti, M. and Meester, R. (2006). Navigation in small-world networks: a scale free continuum model. J. Appl. Prob. 43, 1173–1180.CrossRefGoogle Scholar
Ganesh, A. J. and Draief, M. (2006). Efficient routing in Poisson small world networks. J. Appl. Prob. 43, 678–686.Google Scholar
Ghate, A. (2006). Markov chains, game theory and infinite programming: three paradigms for optimization of complex systems. , The University of Michigan.Google Scholar
Ghate, A. and Smith, R. L. (2008). A hit-and-run approach to generating scale invariant small world networks. To appear in Networks.Google Scholar
Kleinberg, J. M. (2000). Navigation in a small world. Nature 406, 845.Google Scholar
Kleinberg, J. M. (2000). The small world phenomenon: an algorithmic perspective. In Proc. 32nd Annual ACM Symp. Theory Comput. ACM, New York, pp. 163–170.Google Scholar
Liben-Nowell, D. et al. (2005). Geographic routing in social networks. Proc. Nat. Acad. Sci. 102, 11623–11628.Google Scholar
Milgram, S. (1967). The small world problem. Psychology Today 2, 60–67.Google Scholar
Newman, M. E. J. (2000). Models of the small world. J. Statist. Phys. 101, 819–941.Google Scholar
Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45, 167–256.Google Scholar
Travers, J. and Milgram, S. (1969). An experimental study of the small world problem. Sociometry 32, 425–443.Google Scholar
Watts, D. and Strogatz, S. (1998). Collective dynamics of small world networks. Nature 393, 440–442.CrossRefGoogle ScholarPubMed
Watts, D., Dodds, P. S. and Newman, M. E. J. (2002). Identity and search in social networks. Science 296, 1302–1305.CrossRefGoogle ScholarPubMed