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# Phase Transitions of Best-of-Two and Best-of-Three on Stochastic Block Models

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LIPIcs.DISC.2019.32.pdf
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## Acknowledgements

We would like to thank Colin Cooper, Nan Kang and Tomasz Radzik for helpful discussions. We also thank the anonymous reviewers for their helpful comments.

## Cite As

Nobutaka Shimizu and Takeharu Shiraga. Phase Transitions of Best-of-Two and Best-of-Three on Stochastic Block Models. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.DISC.2019.32

## Abstract

This paper is concerned with voting processes on graphs where each vertex holds one of two different opinions. In particular, we study the Best-of-two and the Best-of-three. Here at each synchronous and discrete time step, each vertex updates its opinion to match the majority among the opinions of two random neighbors and itself (the Best-of-two) or the opinions of three random neighbors (the Best-of-three). Previous studies have explored these processes on complete graphs and expander graphs, but we understand significantly less about their properties on graphs with more complicated structures. In this paper, we study the Best-of-two and the Best-of-three on the stochastic block model G(2n,p,q), which is a random graph consisting of two distinct Erdős-Rényi graphs G(n,p) joined by random edges with density q <= p. We obtain two main results. First, if p=omega(log n/n) and r=q/p is a constant, we show that there is a phase transition in r with threshold r^* (specifically, r^*=sqrt{5}-2 for the Best-of-two, and r^*=1/7 for the Best-of-three). If r>r^*, the process reaches consensus within O(log log n+log n/log (np)) steps for any initial opinion configuration with a bias of Omega(n). By contrast, if r<r^*, then there exists an initial opinion configuration with a bias of Omega(n) from which the process requires at least 2^{Omega(n)} steps to reach consensus. Second, if p is a constant and r>r^*, we show that, for any initial opinion configuration, the process reaches consensus within O(log n) steps. To the best of our knowledge, this is the first result concerning multiple-choice voting for arbitrary initial opinion configurations on non-complete graphs.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Stochastic processes
• Mathematics of computing → Random graphs
##### Keywords
• Distributed Voting
• Consensus Problem
• Random Graph

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## References

1. E. Abbe. Community detection and stochastic block models: recent developments. Journal of Machine Learning Research, 18(177):1-86, 2018.
2. E. Abbe and C. Sandon. Recovering communities in the general stochastic block model without knowing the parameters. In Proceedings of the 28th International Conference on Neural Information Processing Systems (NIPS), 1:676-684, 2015.
3. M. A. Abdullah and M. Draief. Global majority consensus by local majority polling on graphs of a given degree sequence. Discrete Applied Mathematics, 1(10):1-10, 2015.
4. Y. Afek, N. Alon, O. Barad, E. Hornstein, N. Barkai, and Z. Bar-Joseph. A biological solution to a fundamental distributed computing problem. Science, 331(6014):183-185, 2011.
5. P. Barbillon, S. Donnet, E. Lazega, and A. Bar-Hen. Stochastic block models for multiplex networks: an application to a multilevel network of researchers. Journal of the Royal Statistical Society Series A, 180(1):295-314, 2017.
6. L. Becchetti, A. Clementi, P. Manurangsi, E. Natale, F. Pasquale, P. Raghavendra, and L. Trevisan. Average whenever you meet: Opportunistic protocols for community detection. In Proceedings of the 26th Annual European Symposium on Algorithms (ESA), 7:1-13, 2018.
7. L. Becchetti, A. Clementi, E. Natale, F. Pasquale, R. Silvestri, and L. Trevisan. Simple dynamics for plurality consensus. Distributed Computing, 30(4):293-306, 2017.
8. L. Becchetti, A. Clementi, E. Natale, F. Pasquale, and L. Trevisan. Stabilizing consensus with many opinions. In Proceedings of the 27th annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 620-635, 2016.
9. L. Becchetti, A. Clementi, E. Natale, F. Pasquale, and L. Trevisan. Find your place: Simple distributed algorithms for community detection. In Proceedings of the 28th annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 940-959, 2017.
10. P. Berenbrink, A. Clementi, R. Elsässer, P. Kling, F. Mallmann-Trenn, and E. Natale. Ignore or comply? On breaking symmetry in consensus. In Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), pages 335-344, 2017.
11. J. Chen and B. Yuan. Detecting functional modules in the yeast protein-protein interaction network. Bioinformatics, 22(18):2283-2290, 2006.
12. C. Cooper, R. Elsässer, H. Ono, and T. Radzik. Coalescing random walks and voting on connected graphs. SIAM Journal on Discrete Mathematics, 27(4):1748-1758, 2013.
13. C. Cooper, R. Elsässer, and T. Radzik. The power of two choices in distributed voting. In Proceedings of the 41st International Colloquium on Automata, Languages, and Programming (ICALP), 2:435-446, 2014.
14. C. Cooper, R. Elsässer, T. Radzik, N. Rivera, and T. Shiraga. Fast consensus for voting on general expander graphs. In Proceedings of the 29th International Symposium on Distributed Computing (DISC), pages 248-262, 2015.
15. C. Cooper, T. Radzik, N. Rivera, and T. Shiraga. Fast plurality consensus in regular expanders. In Proceedings of the 31st International Symposium on Distributed Computing (DISC), 91(13):1-16, 2017.
16. C. Cooper and N. Rivera. The linear voting model. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP), 55(144):1-12, 2016.
17. E. Cruciani, E. Natale, A. Nusser, and G. Scornavacca. Phase transition of the 2-choices dynamics on core-periphery networks. In Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems (AAMAS), pages 777-785, 2018.
18. E. Cruciani, E. Natale, and G. Scornavacca. Distributed community detection via metastability of the 2-choices dynamics. In Proceedings of the 33rd AAAI conference on artificial intelligence (AAAI), pages 6046-6053, 2019.
19. B. Doerr, L. A. Goldberg, L. Minder, T. Sauerwald, and C. Scheideler. Stabilizing consensus with the power of two choices. In Proceedings of the 23rd annual ACM symposium on Parallelism in algorithms and architectures (SPAA), pages 149-158, 2011.
20. M. Fischer, N. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1(1):26-39, 1986.
21. A. Frieze and M. Karońsky. Introduction to random graphs. Campridge University Press, 2016.
22. M. Ghaffari and J. Lengler. Nearly-tight analysis for 2-choice and 3-majority consensus dynamics. In Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), pages 305-313, 2018.
23. S. Gilbert and D. Kowalski. Distributed agreement with optimal communication complexity. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 965-977, 2010.
24. A. Goldenberg, A. X. Zheng, S. E. Fienberg, and E. M. Airoldi. A survey of statistical network models. Foundations and Trends in Machine Learning, 2(2):129-233, 2010.
25. Y. Hassin and D. Peleg. Distributed probabilistic polling and applications to proportionate agreement. Information and Computation, 171(2):248-268, 2001.
26. M. Hirsch and H. L. Smith. Monotone dynamical systems. In Handbook of Differential Equations: Ordinary Differential Equations, 2(4):239-357, 2005.
27. V. Kanade, F. Mallmann-Trenn, and T. Sauerwald. On coalescence time in graphs: When is coalescing as fast as meeting? In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 956-965, 2019.
28. N. Kang and R. Rivera. Best-of-Three voting on dense graphs. In Proceedings of the 31st ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 115-121, 2019.
29. J. H. Kim and V. H. Vu. Concentration of multivariate polynomials and its applications. Combinatorica, 20:417-434, 2000.
30. T. M. Liggett. Interacting particle systems. Springer-Verlag, 1985.
31. E. M. Marcotte, M. Pellegrini, H.-L. Ng, D. W. Rice, T. O. Yeates, and D. Eisenberg. Detecting protein function and protein-protein interactions from genome sequences. Science, 285(5428):751-753, 1999.
32. E. Mossel, J. Neeman, and O. Tamuz. Majority dynamics and aggregation of information in social networks. Autonomous Agents and Multi-Agent Systems, 28(3):408-429, 2014.
33. T. Nakata, H. Imahayashi, and M. Yamashita. Probabilistic local majority voting for the agreement problem on finite graph. In Proceedings of the 5th Annual International Computing and Combinatorics Conference (COCOON), pages 330-338, 1999.
34. R. I. Oliveira and Y. Peres. Random walks on graphs: new bounds on hitting, meeting, coalescing and returning. In Proceedings of the 16th Workshop on Analytic Algorithmics and Combinatorics (ANALCO), pages 119-126, 2019.
35. G. Schoenebeck and F. Yu. Consensus of interacting particle systems on Erdős-Rényi graphs. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1945-1964, 2018.
36. N. Shimizu and T. Shiraga. Phase Transitions of Best-of-Two and Best-of-Three on Stochastic Block Models. arXiv, 2019. URL: http://arxiv.org/abs/1907.12212.