## Bernoulli

• Bernoulli
• Volume 19, Number 4 (2013), 1122-1149.

### Interacting particle systems as stochastic social dynamics

David Aldous

#### Abstract

The style of mathematical models known to probabilists as Interacting Particle Systems and exemplified by the Voter, Exclusion and Contact processes have found use in many academic disciplines. In many such disciplines the underlying conceptual picture is of a social network, where individuals meet pairwise and update their “state” (opinion, activity etc) in a way depending on the two previous states. This picture motivates a precise general setup we call Finite Markov Information Exchange (FMIE) processes. We briefly describe a few less familiar models (Averaging, Compulsive Gambler, Deference, Fashionista) suggested by the social network picture, as well as a few familiar ones.

#### Article information

Source
Bernoulli, Volume 19, Number 4 (2013), 1122-1149.

Dates
First available in Project Euclid: 27 August 2013

https://projecteuclid.org/euclid.bj/1377612846

Digital Object Identifier
doi:10.3150/12-BEJSP04

Mathematical Reviews number (MathSciNet)
MR3102546

Zentralblatt MATH identifier
1278.60141

#### Citation

Aldous, David. Interacting particle systems as stochastic social dynamics. Bernoulli 19 (2013), no. 4, 1122--1149. doi:10.3150/12-BEJSP04. https://projecteuclid.org/euclid.bj/1377612846

#### References

• [1] Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Applied Mathematical Sciences 77. New York: Springer.
• [2] Aldous, D. (2010). When knowing early matters: Gossip, percolation and Nash equilibria. Available at arXiv:1005.4846.
• [3] Aldous, D. (2012). Finite Markov information-exchange processes. Beamer slides of five lectures. Available at http://www.stat.berkeley.edu/~aldous/FMIE/2012_FMIE_lectures.html.
• [4] Aldous, D. and Fill, J. (2000). Reversible Markov chains and random walks on graphs. Available at http://www.stat.berkeley.edu/~aldous/RWG/book.html.
• [5] Aldous, D. and Lanoue, D. (2012). A lecture on the averaging process. Probab. Surv. 9 90–102.
• [6] Aldous, D. and Steele, J.M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 1–72. Berlin: Springer.
• [7] Aldous, D.J. (1982). Some inequalities for reversible Markov chains. J. London Math. Soc. (2) 25 564–576.
• [8] Aldous, D.J. (2003). A stochastic complex network model. Electron. Res. Announc. Amer. Math. Soc. 9 152–161 (electronic).
• [9] Aldous, D.J. and Shun, J. (2010). Connected spatial networks over random points and a route-length statistic. Statist. Sci. 25 275–288.
• [10] Bailey, N.T.J. (1950). A simple stochastic epidemic. Biometrika 37 193–202.
• [11] Bailey, N.T.J. (1957). The Mathematical Theory of Epidemics. New York: Hafner Publishing Co.
• [12] Benaïm, M. and Rossignol, R. (2008). Exponential concentration for first passage percolation through modified Poincaré inequalities. Ann. Inst. Henri Poincaré Probab. Stat. 44 544–573.
• [13] Berestycki, N. (2009). Recent Progress in Coalescent Theory. Ensaios Matemáticos [Mathematical Surveys] 16. Rio de Janeiro: Sociedade Brasileira de Matemática.
• [14] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2010). First passage percolation on random graphs with finite mean degrees. Ann. Appl. Probab. 20 1907–1965.
• [15] Blythe, R.A. (2010). Ordering in voter models on networks: Exact reduction to a single-coordinate diffusion. J. Phys. A 43 385003, 33.
• [16] Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge: Cambridge Univ. Press.
• [17] Bordenave, C., Gousseau, Y. and Roueff, F. (2006). The dead leaves model: A general tessellation modeling occlusion. Adv. in Appl. Probab. 38 31–46.
• [18] Caputo, P., Liggett, T.M. and Richthammer, T. (2010). Proof of Aldous’ spectral gap conjecture. J. Amer. Math. Soc. 23 831–851.
• [19] Castellano, C., Fortunato, S. and Loreto, V. (2009). Statistical physics of social dynamics. Rev. Mod. Phys. 81 591–646.
• [20] Chatterjee, S. and Durrett, R. (2011). Asymptotic behavior of Aldous’ gossip process. Ann. Appl. Probab. 21 2447–2482.
• [21] Chierichetti, F., Lattanzi, S. and Panconesi, A. (2010). Rumour spreading and graph conductance. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms 1657–1663. Philadelphia, PA: SIAM.
• [22] Cooper, C., Elsässer, R., Ono, H. and Radzik, T. (2012). Coalescing random walks and voting on graphs. In Proceedings of the 2012 ACM symposium on Principles of Distributed Computing 37–56. ACM: New York.
• [23] Cox, J.T. (1989). Coalescing random walks and voter model consensus times on the torus in $\mathbf{Z}^{d}$. Ann. Probab. 17 1333–1366.
• [24] Cox, J.T. and Griffeath, D. (1986). Diffusive clustering in the two-dimensional voter model. Ann. Probab. 14 347–370.
• [25] Diaconis, P. (1996). The cutoff phenomenon in finite Markov chains. Proc. Natl. Acad. Sci. USA 93 1659–1664.
• [26] Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695–750.
• [27] Dorogovtsev, S.N. and Mendes, J.F.F. (2003). Evolution of Networks. Oxford: Oxford Univ. Press.
• [28] Durrett, R. (1991). The contact process, 1974–1989. In Mathematics of Random Media (Blacksburg, VA, 1989). Lectures in Applied Mathematics 27 1–18. Providence, RI: Amer. Math. Soc.
• [29] Durrett, R. (2008). Probability Models for DNA Sequence Evolution, 2nd ed. Probability and Its Applications (New York). New York: Springer.
• [30] Durrett, R. (2010). Some features of the spread of epidemics and opinions on a random graph. Proc. Natl. Acad. Sci. USA 107 4491–4498.
• [31] Durrett, R. and Schonmann, R.H. (1988). The contact process on a finite set. II. Ann. Probab. 16 1570–1583.
• [32] Fill, J.A. and Pemantle, R. (1993). Percolation, first-passage percolation and covering times for Richardson’s model on the $n$-cube. Ann. Appl. Probab. 3 593–629.
• [33] Floridi, L. (2010). Information: A Very Short Introduction. Oxford, UK: Oxford Univ. Press.
• [34] Griffeath, D. (1979). Additive and Cancellative Interacting Particle Systems. Lecture Notes in Math. 724. Berlin: Springer.
• [35] Kesten, H. (2003). First-passage percolation. In From Classical to Modern Probability. Progress in Probability 54 93–143. Basel: Birkhäuser.
• [36] Levin, D.A., Peres, Y. and Wilmer, E.L. (2009). Markov Chains and Mixing Times. Providence, RI: Amer. Math. Soc. With a chapter by James G. Propp and David B. Wilson.
• [37] Liggett, T.M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. New York: Springer.
• [38] Montenegro, R. and Tetali, P. (2006). Mathematical aspects of mixing times in Markov chains. Found. Trends Theor. Comput. Sci. 1 1–121.
• [39] Mountford, T., Mourrat, J.-C., Valesin, D. and Yao, Q. (2012). Exponential extinction time of the contact process on finite graphs. Available at arXiv:1203.2972.
• [40] Newman, M.E.J. (2003). The structure and function of complex networks. SIAM Rev. 45 167–256 (electronic).
• [41] Oliveira, R.I. (2012). On the coalescence time of reversible random walks. Trans. Amer. Math. Soc. 364 2109–2128.
• [42] Oliveira, R.I. (2012). Mean field conditions for coalescing random walks. Available at arXiv:1109.5684.
• [43] Shah, D. (2008). Gossip algorithms. Foundations and Trends in Networking 3 1–125.
• [44] Social Cognitive Networks Academic Research Center (2011). Minority rules: Scientists discover tipping point for the spread of ideas. Available at http://scnarc.rpi.edu/content/minority-rules-scientists-discover-tipping-point-spread-ideas.
• [45] van der Hofstad, R., Hooghiemstra, G. and Van Mieghem, P. (2002). The flooding time in random graphs. Extremes 5 111–129 (2003).
• [46] Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge: Cambridge Univ. Press.
• [47] Zähle, I., Cox, J.T. and Durrett, R. (2005). The stepping stone model. II. Genealogies and the infinite sites model. Ann. Appl. Probab. 15 671–699.