</script> To interpret interacting particle system style models as social dynamics, suppose each pair ({i,j}) of individuals in a finite population meet at random times of arbitrary specified rates (nu_{ij}), and update their states according to some specified rule. The <i>averaging process</i> has real-valued states and the rule: upon meeting, the values (X_i(t-), X_j(t-)) are replaced by (frac{1}{2}(X_i(t-)+X_j(t-)), frac{1}{2}(X_i(t-)+X_j(t-))). It is curious this simple process has not been studied very systematically. We provide an expository account of basic facts and open problems."> click here</a> to read the full-text legal code). Under the CCAL, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles in <i>Probability Surveys</i>, so long as the original authors and source are credited. This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your work will ensure your right to make your work freely and openly available. <b>Summary of the Creative Commons Attribution License</b> You are free: <ul> <li>to copy, distribute, display, and perform the work <li>to make derivative works <li>to make commercial use of the work </ul> under the following condition of Attribution: others must attribute the work if displayed on the web or stored in any electronic archive by making a link back to the website of the Journal via its Digital Object Identifier (DOI), or if published in other media by acknowledging prior publication in this Journal with a precise citation including the DOI. For any further reuse or distribution, the same terms apply. Any of these conditions can be waived by permission of The Corresponding Author. We are using the same license as the <a href=http://www.i-journals.org/ps/"http://www.plos.org/">Public Library of Science</a> (PLoS). ">
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 Probability Surveys > Vol. 9 (2012) open journal systems 


A lecture on the averaging process

David Aldous, University of California, Berkeley
Daniel Lanoue, University of California, Berkeley


Abstract
To interpret interacting particle system style models as social dynamics, suppose each pair \(\{i,j\}\) of individuals in a finite population meet at random times of arbitrary specified rates \(\nu_{ij}\), and update their states according to some specified rule. The averaging process has real-valued states and the rule: upon meeting, the values \(X_i(t-), X_j(t-)\) are replaced by \(\frac{1}{2}(X_i(t-)+X_j(t-)), \frac{1}{2}(X_i(t-)+X_j(t-))\). It is curious this simple process has not been studied very systematically. We provide an expository account of basic facts and open problems.

AMS 2000 subject classifications: Primary 60K35; secondary 60K99.

Keywords: Duality, interacting particle systems, rate of convergence, spectral gap, voter model.

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Aldous, David, Lanoue, Daniel, A lecture on the averaging process, Probability Surveys, 9, (2012), 90-102 (electronic). DOI: 10.1214/11-PS184.

References

[1]    Acemoglu, D., Como, G., Fagnini, F. and Ozdaglar, A. (2011). Opinion fluctuations and disagreement in social networks. http://arxiv.org/abs/1009.2653

[2]    Aldous, D. (2011). Finite Markov Information-Exchange Procesess. Lecture Notes from Spring 2011. http://www.stat.berkeley.edu/~aldous/260-FMIE/Lectures/index.html.

[3]     Aldous, D. and Fill, J. Reversible Markov Chains and Random Walks on Graphs. http://www.stat.berkeley.edu/~aldous/RWG/book.html

[4]    Ben-Naim, E., Krapivsky, P.L. and Redner, S. (2003). Bifurcations and patterns in compromise processes. Phys. D 183 190–204.

[5]    Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695–750. MR1410112

[6]    Häggström, O. (2011). A pairwise averaging procedure with application to consensus formation in the Deffuant model. http://www.math.chalmers.se/~olleh/averaging.pdf.

[7]    Lanchier, N. (2011). The critical value of the bounded confidence Deffuant model equals one half. http://stat.asu.edu/~lanchier/articles/2011h_lanchier.pdf.

[8]    Levin, D. A., Peres, Y. and Wilmer, E. L. (2009), Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI. MR2466937

[9]    Liggett, T. M. (1985). Interacting Particle Systems. Springer-Verlag, New York. MR776231

[10]    Montenegro, R. and Tetali, P. (2006). Mathematical aspects of mixing times in Markov chains. Found. Trends Theor. Comput. Sci. 1 1–121. MR2341319

[11]    Olshevsky, A. and Tsitsiklis, J. N. (2009). Convergence speed in distributed consensus and averaging. SIAM J. Control Optim. 48 33–55. MR2480125

[12]    Shah, D. (2008). Gossip algorithms. Foundations and Trends in Networking 3 1–125.




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Probability Surveys. ISSN: 1549-5787