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Asynchronous Majority Dynamics on Binomial Random Graphs

Authors Divyarthi Mohan , Paweł Prałat

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ACM Subject Classification
  • Mathematics of computing → Stochastic processes
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Social networks
Keywords
  • Opinion dynamics
  • Social learning
  • Stochastic processes
  • Random Graphs
  • Consensus

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Abstract

We study information aggregation in networks when agents interact to learn a binary state of the world. Initially each agent privately observes an independent signal which is correct with probability 1/2+δ for some δ > 0. At each round, a node is selected uniformly at random to update their public opinion to match the majority of their neighbours (breaking ties in favour of their initial private signal). Our main result shows that for sparse and connected binomial random graphs G(n,p) the process stabilizes in a correct consensus in 𝒪(nlog² n/log log n) steps with high probability. In fact, when log n/n ≪ p = o(1) the process terminates at time T^ = (1+o(1))nlog n, where T^ is the first time when all nodes have been selected at least once. However, in dense binomial random graphs with p = Ω(1), there is an information cascade where the process terminates in the incorrect consensus with probability bounded away from zero.

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Divyarthi Mohan and Paweł Prałat. Asynchronous Majority Dynamics on Binomial Random Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.5

Author Details

Divyarthi Mohan
  • Blavatnik School of Computer Science, Tel Aviv University, Israel
Paweł Prałat
  • Department of Mathematics, Toronto Metropolitan University, Canada

Funding

  • Mohan, Divyarthi: This project was funded in part by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement no. 866132) and by the Israel Science Foundation (grant no. 317/17).

Acknowledgements

This work was done while the authors were visiting the Simons Institute for the Theory of Computing.

References

  1. Mohammed Amin Abdullah and Moez Draief. Global majority consensus by local majority polling on graphs of a given degree sequence. Discrete Applied Mathematics, 180:1-10, 2015. Google Scholar
  2. Daron Acemoglu, Munther A. Dahleh, Ilan Lobel, and Asuman Ozdaglar. Bayesian learning in social networks. Review of Economic Studies, 78(4):1201-1236, 2011. Google Scholar
  3. Aris Anagnostopoulos, Luca Becchetti, Emilio Cruciani, Francesco Pasquale, and Sara Rizzo. Biased opinion dynamics: when the devil is in the details. Information Sciences, 593:49-63, 2022. Google Scholar
  4. Maryam Bahrani, Nicole Immorlica, Divyarthi Mohan, and S Matthew Weinberg. Asynchronous majority dynamics in preferential attachment trees. 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), 2020. Google Scholar
  5. Abhijit Banerjee, Emily Breza, Arun G Chandrasekhar, and Markus Mobius. Naive learning with uninformed agents. American Economic Review, 111(11):3540-3574, 2021. Google Scholar
  6. Abhijit Banerjee and Drew Fudenberg. Word-of-mouth learning. Games and Economic Behavior, 46(1):1-22, January 2004. URL: http://ideas.repec.org/a/eee/gamebe/v46y2004i1p1-22.html.
  7. Abhijit V. Banerjee. A simple model of herd behavior. The Quarterly Journal of Economics, 107(3):797-817, 1992. Google Scholar
  8. Luca Becchetti, Andrea Clementi, and Emanuele Natale. Consensus dynamics: An overview. SIGACT News, 51(1):58-104, March 2020. URL: https://doi.org/10.1145/3388392.3388403.
  9. Luca Bechetti, Andrea E.F. Clementi, Emanuele Natale, Francesco Pasquale, and Luca Trevisan. Stabilizing consensus with many opinions. In ACM Symposium on Discrete Algorithms (SODA), 2016. Google Scholar
  10. Itai Benjamini, Siu-On Chan, Ryan O’Donnell, Omer Tamuz, and Li-Yang Tan. Convergence, unanimity and disagreement in majority dynamics on unimodular graphs and random graphs. Stochastic Processes and their Applications, 126(9):2719-2733, 2016. Google Scholar
  11. Sushil Bikhchandani, David Hirshleifer, Omer Tamuz, and Ivo Welch. Information cascades and social learning. Technical report, National Bureau of Economic Research, 2021. Google Scholar
  12. Sushil Bikhchandani, David Hirshleifer, and Ivo Welch. A theory of fads, fashion, custom, and cultural change in informational cascades. Journal of Political Economy, 100(5):992-1026, October 1992. Google Scholar
  13. Béla Bollobás. Random graphs. Cambridge University Press, 2001. Google Scholar
  14. Boğaçhan Çelen and Shachar Kariv. Observational learning under imperfect information. Games and Economic behavior, 47(1):72-86, 2004. Google Scholar
  15. Debsoumya Chakraborti, Jeong Han Kim, Joonkyung Lee, and Tuan Tran. Majority dynamics on sparse random graphs. Random Structures & Algorithms, 63(1):171-191, 2023. Google Scholar
  16. Fan RK Chung and Linyuan Lu. Complex graphs and networks. American Mathematical Soc., 2006. Google Scholar
  17. Peter Clifford and Aidan Sudbury. A model for spatial conflict. Biometrika, 60(3):581-588, 1973. URL: http://www.jstor.org/stable/2335008.
  18. Emilio Cruciani, Hlafo Alfie Mimun, Matteo Quattropani, and Sara Rizzo. Phase transitions of the k-majority dynamics in a biased communication model. In Proceedings of the 22nd International Conference on Distributed Computing and Networking, pages 146-155, 2021. Google Scholar
  19. Morris H. DeGroot. Reaching a consensus. Review of Economic Studies, 69(345):118-121, 1974. Google Scholar
  20. Michal Feldman, Nicole Immorlica, Brendan Lucier, and S. Matthew Weinberg. Reaching consensus via non-bayesian asynchronous learning in social networks. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2014, 2014. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.192.
  21. Nikolaos Fountoulakis, Mihyun Kang, and Tamás Makai. Resolution of a conjecture on majority dynamics: Rapid stabilization in dense random graphs. Random Structures & Algorithms, 57(4):1134-1156, 2020. Google Scholar
  22. Douglas Gale and Shachar Kariv. Bayesian learning in social networks. Games and Economic Behavior, 45(2):329-346, 2003. Special Issue in Honor of Robert W. Rosenthal. URL: https://doi.org/10.1016/S0899-8256(03)00144-1.
  23. Bernd Gärtner and Ahad N Zehmakan. Majority model on random regular graphs. In LATIN 2018: Theoretical Informatics: 13th Latin American Symposium, Buenos Aires, Argentina, April 16-19, 2018, Proceedings 13, pages 572-583. Springer, 2018. Google Scholar
  24. Mohsen Ghaffari and Johannes Lengler. Nearly-tight analysis for 2-choice and 3-majority consensus dynamics. In ACM Symposium on Principles of Distributed Computing (PODC), 2018. Google Scholar
  25. Mohsen Ghaffari and Merav Parter. A polylogarithmic gossip algorithm for plurality consensus. In ACM Symposium on Principles of Distributed Computing (PODC), 2016. Google Scholar
  26. Benjamin Golub and Matthew O. Jackson. Naïve learning in social networks and the wisdom of crowds. American Economic Journal: Microeconomics, 2(1):112-149, 2010. Google Scholar
  27. Richard A. Holley and Thomas M. Liggett. Ergodic theorems for weakly interacting infinite systems and the voter model. The Annals of Probability, 3(4):643-663, 1975. URL: http://www.jstor.org/stable/2959329.
  28. C Douglas Howard. Zero-temperature ising spin dynamics on the homogeneous tree of degree three. Journal of applied probability, 37(3):736-747, 2000. Google Scholar
  29. Svante Janson, Tomasz Łuczak, and Andrzej Ruciński. Random graphs, volume 45. John Wiley & Sons, 2011. Google Scholar
  30. Bogumił Kamiński, Paweł Prałat, and François Théberge. Artificial benchmark for community detection (abcd)—fast random graph model with community structure. Network Science, 9(2):153-178, 2021. Google Scholar
  31. Y. Kanoria and O. Tamuz. Tractable bayesian social learning on trees. IEEE Journal on Selected Areas in Communications, 31(4):756-765, 2013. Google Scholar
  32. Yashodhan Kanoria, Andrea Montanari, et al. Majority dynamics on trees and the dynamic cavity method. The Annals of Applied Probability, 21(5):1694-1748, 2011. Google Scholar
  33. Michal Karoński and Alan Frieze. Introduction to Random Graphs. Cambridge University Press, 2016. Google Scholar
  34. Marcos Kiwi, Lyuben Lichev, Dieter Mitsche, and Paweł Prałat. Label propagation on binomial random graphs. arXiv preprint, 2023. URL: https://arxiv.org/abs/2302.03569.
  35. S. Matwin, A. Milios, P. Prałat, A. Soares, and F. Théberge. Generative Methods for Social Media Analysis. SpringerBriefs in Computer Science. Springer Nature Switzerland, 2023. URL: https://books.google.ca/books?id=wPbJEAAAQBAJ.
  36. Colin McDiarmid. Concentration. In Probabilistic methods for algorithmic discrete mathematics, pages 195-248. Springer, 1998. Google Scholar
  37. Markus Mobius and Tanya Rosenblat. Social learning in economics. Annual Review of Economics, 6(1):827-847, 2014. URL: https://doi.org/10.1146/annurev-economics-120213-012609.
  38. E. Mossel, N. Olsman, and O. Tamuz. Efficient bayesian learning in social networks with gaussian estimators. In 2016 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton), 2016. Google Scholar
  39. Elchanan Mossel, Joe Neeman, and Omer Tamuz. Majority dynamics and aggregation of information in social networks. In Autonomous Agents and Multi-Agent Systems (AAMAS), 2013. Google Scholar
  40. Elchanan Mossel, Allan Sly, and Omer Tamuz. Asymptotic learning on bayesian social networks. Probability Theory and Related Fields, 2014. Google Scholar
  41. Manuel Mueller-Frank. A general framework for rational learning in social networks. Theoretical Economics, 8(1):1-40, 2013. Google Scholar
  42. Dinah Rosenberg, Eilon Solan, and Nicolas Vieille. Informational externalities and emergence of consensus. Games and Economic Behavior, 66(2):979-994, 2009. Google Scholar
  43. Lones Smith and Peter Sorensen. Pathological outcomes of observational learning. Econometrica, 68(2):371-398, March 2000. URL: http://ideas.repec.org/a/ecm/emetrp/v68y2000i2p371-398.html.
  44. Omer Tamuz and Ran Tessler. Majority dynamics and the retention of information. In Working paper, 2013. Google Scholar
  45. Linh Tran and Van Vu. Reaching a consensus on random networks: The power of few. arXiv preprint, 2019. URL: https://arxiv.org/abs/1911.10279.
  46. Linh Tran and Van Vu. The "power of few" phenomenon: The sparse case. arXiv preprint, 2023. URL: https://arxiv.org/abs/2302.05605.
  47. Ahad N Zehmakan. Opinion forming in Erdős-Rényi random graph and expanders. Discrete Applied Mathematics, 277:280-290, 2020. Google Scholar
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