# Majority dynamics and aggregation of information in social networks

- 545 Downloads
- 10 Citations

## Abstract

Consider \(n\) individuals who, by popular vote, choose among \(q \ge 2\) alternatives, one of which is “better” than the others. Assume that each individual votes independently at random, and that the probability of voting for the better alternative is larger than the probability of voting for any other. It follows from the law of large numbers that a plurality vote among the \(n\) individuals would result in the correct outcome, with probability approaching one exponentially quickly as \(n \rightarrow \infty \). Our interest in this article is in a variant of the process above where, after forming their initial opinions, the voters update their decisions based on some interaction with their neighbors in a social network. Our main example is “majority dynamics”, in which each voter adopts the most popular opinion among its friends. The interaction repeats for some number of rounds and is then followed by a population-wide plurality vote. The question we tackle is that of “efficient aggregation of information”: in which cases is the better alternative chosen with probability approaching one as \(n \rightarrow \infty \)? Conversely, for which sequences of growing graphs does aggregation fail, so that the wrong alternative gets chosen with probability bounded away from zero? We construct a family of examples in which interaction prevents efficient aggregation of information, and give a condition on the social network which ensures that aggregation occurs. For the case of majority dynamics we also investigate the question of unanimity in the limit. In particular, if the voters’ social network is an expander graph, we show that if the initial population is sufficiently biased towards a particular alternative then that alternative will eventually become the unanimous preference of the entire population.

## Keywords

Social networks Aggregation of information Majority dynamics Discrete Fourier analysis## Notes

### Acknowledgments

We would like to thank Miklos Racz for his careful reading of the manuscript and his suggestions. Elchanan Mossel is supported by a Sloan fellowship in Mathematics, by BSF Grant 2004105, by NSF Career Award (DMS 054829), by ONR Award N00014-07-1-0506 and by ISF Grant 1300/08. Omer Tamuz is supported by ISF Grant 1300/08. Omer Tamuz is a recipient of the Google Europe Fellowship in Social Computing, and this research is supported in part by this Google Fellowship.

## References

- 1.Alon, N., & Spencer, J. (2008).
*The probabilistic method*(Vol. 73). New York: Wiley-Interscience.Google Scholar - 2.Bala, V., & Goyal, S. (1998). Learning from neighbours.
*Review of Economic Studies*,*65*(3), 595–621.CrossRefzbMATHGoogle Scholar - 3.Bawa, M., Garcia-Molina, H., Gionis, A., & Motwani R. (2003).
*Estimating aggregates on a peer-to-peer network*. submitted for publication.Google Scholar - 4.Berger, E. (2001). Dynamic monopolies of constant size.
*Journal of Combinatorial Theory, Series B*,*83*(2), 191–200.CrossRefzbMATHMathSciNetGoogle Scholar - 5.Condorcet, J.-A.-N. (1785). Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix. De l’Imprimerie Royale.Google Scholar
- 6.DeGroot, M. H. (1974). Reaching a consensus.
*Journal of the American Statistical Association*,*69*(345), 118–121.CrossRefzbMATHGoogle Scholar - 7.Efron, B., & Stein, C. (1981). The jackknife estimate of variance.
*The Annals of Statistics*,*9*(3), 586–596.CrossRefzbMATHMathSciNetGoogle Scholar - 8.Fontes, L., Schonmann, R., & Sidoravicius, V. (2002). Stretched exponential fixation in stochastic ising models at zero temperature.
*Communications in Mathematical Physics*,*228*(3), 495–518.CrossRefzbMATHMathSciNetGoogle Scholar - 9.Friedgut, E., & Kalai, G. (1996). Every monotone graph property has a sharp threshold.
*Proceedings of the American Mathematical Society*,*124*(10), 2993–3002.CrossRefzbMATHMathSciNetGoogle Scholar - 10.Goles, E., & Olivos, J. (1980). Periodic behaviour of generalized threshold functions.
*Discrete Mathematics*,*30*(2), 187–189.CrossRefzbMATHMathSciNetGoogle Scholar - 11.Golub, B., & Jackson, M. (2010). Naive learning in social networks and the wisdom of crowds.
*American Economic Journal: Microeconomics*,*2*(1), 112–149.Google Scholar - 12.Hoory, S., Linial, N., & Wigderson, A. (2006). Expander graphs and their applications.
*Bulletin of the American Mathematical Society*,*43*(4), 439–561.CrossRefzbMATHMathSciNetGoogle Scholar - 13.Howard, C. (2000). Zero-temperature ising spin dynamics on the homogeneous tree of degree three.
*Journal of Applied Probability*,*37*, 736–747.CrossRefzbMATHMathSciNetGoogle Scholar - 14.Kahn, J., Kalai, G., & Linial, N. (1988). The influence of variables on boolean functions. In
*Proceedings of the 29th Annual Symposium on Foundations of Computer Science*(pp. 68–80).Google Scholar - 15.Kalai, G. (2001).
*Social choice and threshold phenomena*. Discussion Paper Series.Google Scholar - 16.Kalai, G. (2004). Social indeterminacy.
*Econometrica*,*72*, 1565–1581.CrossRefzbMATHMathSciNetGoogle Scholar - 17.Kalai, G., & Mossel, E. (2010).
*Sharp thresholds for non-boolean functions and social choice theory*. Preprint.Google Scholar - 18.Kanoria, Y., & Montanari, A. (2009).
*Majority dynamics on trees and the dynamic cavity method*. Arxiv, preprint arXiv:0907.0449.Google Scholar - 19.Kempe, D., Dobra, A., & Gehrke, J. (2003). Gossip-based computation of aggregate information. In
*Proceedings of the 44th Annual Symposium on Foundations of Computer Science*(pp. 482–491). New York: IEEE.Google Scholar - 20.Margulis, G. (1977). Probabilistic characteristic of graphs with large connectivity.
*Problems of Information Transmission*,*10*, 174–179.Google Scholar - 21.McDiarmid, C. (1989). On the method of bounded differences.
*Surveys in Combinatorics*,*141*(1), 148–188.MathSciNetGoogle Scholar - 22.Mossel, E., Sly, A., & Tamuz, O. (2012).
*Asymptotic learning on Bayesian social networks*. Preprint at http://arxiv.org/abs/1207.5893 - 23.Mossel, E., & Tamuz, O. (2012). Complete characterization of functions satisfying the conditions of arrows theorem.
*Social Choice and Welfare*,*39*(1), 127–140.CrossRefzbMATHMathSciNetGoogle Scholar - 24.Russo, L. (1982). An approximate zero-one law.
*Probability Theory and Related Fields*,*61*(1), 129–139.zbMATHGoogle Scholar - 25.Shah, D. (2009). Gossip algorithms.
*Foundations and Trends in Networking*,*3*(1), 1–125.CrossRefGoogle Scholar - 26.Talagrand, M. (1994). On Russo’s approximate zero-one law.
*The Annals of Probability*,*22*(3), 1576–1587.CrossRefzbMATHMathSciNetGoogle Scholar