Document

# Discrete Incremental Voting

## File

LIPIcs.OPODIS.2023.10.pdf
• Filesize: 0.94 MB
• 22 pages

## Cite As

Colin Cooper, Tomasz Radzik, and Takeharu Shiraga. Discrete Incremental Voting. In 27th International Conference on Principles of Distributed Systems (OPODIS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 286, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.OPODIS.2023.10

## Abstract

We consider a type of pull voting suitable for discrete numeric opinions which can be compared on a linear scale, for example, 1 ("disagree strongly"), 2 ("disagree"), …, 5 ("agree strongly"). On observing the opinion of a random neighbour, a vertex changes its opinion incrementally towards the value of the neighbour’s opinion, if different. For opinions drawn from a set {1,2,…,k}, the opinion of the vertex would change by +1 if the opinion of the neighbour is larger, or by -1, if it is smaller. It is not clear how to predict the outcome of this process, but we observe that the total weight of the system, that is, the sum of the individual opinions of all vertices, is a martingale. This allows us analyse the outcome of the process on some classes of dense expanders such as complete graphs K_n and random graphs G_{n,p} for suitably large p. If the average of the original opinions satisfies i ≤ c ≤ i+1 for some integer i, then the asymptotic probability that opinion i wins is i+1-c, and the probability that opinion i+1 wins is c-i. With high probability, the winning opinion cannot be other than i or i+1. To contrast this, we show that for a path and opinions 0,1,2 arranged initially in non-decreasing order along the path, the outcome is very different. Any of the opinions can win with constant probability, provided that each of the two extreme opinions 0 and 2 is initially supported by a constant fraction of vertices.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Distributed algorithms
##### Keywords
• Random distributed processes
• Pull voting

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## 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. URL: https://doi.org/10.1016/J.DAM.2014.07.026.
2. David Aldous and James Allen Fill. Reversible markov chains and random walks on graphs. Unfinished monograph (recompiled version, 2014), 2002.
3. Luca Becchetti, Andrea Clementi, Emanuele Natale, Francesco Pasquale, and Riccardo Silvestri. Plurality consensus in the gossip model. In Proceedings, 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 371-390, Philadelphia, PA, 2015. SIAM. URL: https://doi.org/10.1137/1.9781611973730.27.
4. Luca Becchetti, Andrea Clementi, Emanuele Natale, Francesco Pasquale, Riccardo Silvestri, and Luca Trevisan. Simple dynamics for plurality consensus. In Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 247-256, New York, NY, USA, 2014. ACM. URL: https://doi.org/10.1145/2612669.2612677.
5. Luca Becchetti, Andrea Clementi, Emanuele Natale, Francesco Pasquale, and Luca Trevisan. Stabilizing consensus with many opinions. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 620-635, Philadelphia, PA, 2016. SIAM. URL: https://doi.org/10.1137/1.9781611974331.CH46.
6. Petra Berenbrink, George Giakkoupis, and Peter Kling. Tight bounds for coalescing-branching random walks on regular graphs. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1715-1733, Philadelphia, PA, 2018. SIAM. URL: https://doi.org/10.1137/1.9781611975031.112.
7. Siddhartha Brahma, Sandeep Macharla, Sudebkumar Prasant Pal, and Sudhir Kumar Singh. Fair leader election by randomized voting. In Proceedings of the 1st international conference on Distributed Computing and Internet Technology (ICDCIT), pages 22-31. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-30555-2_4.
8. Colin Cooper, R. Elsässer, and Tomasz Radzik. The power of two choices in distributed voting. In Proceedings of the 41st International Colloquium on Automata, Languages, and Programming (ICALP), pages 435-446. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-43951-7_37.
9. Colin Cooper, Robert Elsässer, Hirotaka Ono, and Tomasz Radzik. Coalescing random walks and voting on graphs. In Proceedings of the 31st Annual ACM Symposium on Principles of Distributed Computing (PODC), pages 47-56, New York, NY, USA, 2012. ACM. URL: https://doi.org/10.1145/2332432.2332440.
10. Colin Cooper, Tomasz Radzik, Nicolás Rivera, and Takeharu Shiraga. Fast plurality consensus in regular expanders. In Proceedings of the 31st International Symposium on Distributed Computing (DISC), volume 91, pages 13:1-13:16. Springer, 2017. URL: https://doi.org/10.4230/LIPICS.DISC.2017.13.
11. Benjamin Doerr, Leslie Ann Goldberg, Lorenz Minder, Thomas Sauerwald, and Christian Scheideler. Stabilizing consensus with the power of two choices. In Proceedings of the 23rd ACM symposium on Parallelism in algorithms and architectures (SPAA), pages 149-158, New York, NY, USA, 2011. ACM. URL: https://doi.org/10.1145/1989493.1989516.
12. Mohsen Ghaﬀari and Johannes Lengler. Nearly-tight analysis for 2-choice and 3-majority consensus dynamics. In Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing (PODC), pages 305-313, New York, NY, USA, 2018. ACM. URL: https://dl.acm.org/citation.cfm?id=3212738.
13. Yehuda Hassin and David Peleg. Distributed probabilistic polling and applications to proportionate agreement. Information and Computation, 171(2):248-268, 2001. URL: https://doi.org/10.1006/INCO.2001.3088.
14. Barry Johnson. Design and analysis of fault tolerant digital systems. Addison-Wesley, Boston, MA, USA, 1989.
15. Nan Kang and Nicolás 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, New York, NY, USA, 2019. ACM. URL: https://doi.org/10.1145/3323165.3323207.
16. Toshio Nakata, Hiroshi Imahayashi, and Masafumi 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. Springer, 1999. URL: https://doi.org/10.1007/3-540-48686-0_33.
17. Roberto Imbuzeiro Oliveira and Yuval 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, Philadelphia, PA, 2019. SIAM. URL: https://doi.org/10.1137/1.9781611975505.13.
18. Nobutaka Shimizu and Takeharu Shiraga. Phase transitions of best-of-two and best-of-three on stochastic block models. Random Structures and algorithms, 59(1):96-140, 2021. URL: https://doi.org/10.1002/RSA.20992.