Quasi-Majority Functional Voting on Expander Graphs
Consider a distributed graph where each vertex holds one of two distinct opinions. In this paper, we are interested in synchronous voting processes where each vertex updates its opinion according to a predefined common local updating rule. For example, each vertex adopts the majority opinion among 1) itself and two randomly picked neighbors in best-of-two or 2) three randomly picked neighbors in best-of-three. Previous works intensively studied specific rules including best-of-two and best-of-three individually.
In this paper, we generalize and extend previous works of best-of-two and best-of-three on expander graphs by proposing a new model, quasi-majority functional voting. This new model contains best-of-two and best-of-three as special cases. We show that, on expander graphs with sufficiently large initial bias, any quasi-majority functional voting reaches consensus within O(log n) steps with high probability. Moreover, we show that, for any initial opinion configuration, any quasi-majority functional voting on expander graphs with higher expansion (e.g., Erdős-Rényi graph G(n,p) with p = Ω(1/√n)) reaches consensus within O(log n) with high probability. Furthermore, we show that the consensus time is O(log n/log k) of best-of-(2k+1) for k = o(n/log n).
Distributed voting
consensus problem
expander graph
Markov chain
Theory of computation~Random walks and Markov chains
Theory of computation~Distributed algorithms
97:1-97:19
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/2002.07411.
Nobutaka
Shimizu
Nobutaka Shimizu
The University of Tokyo, Japan
JSPS KAKENHI Grant Number 19J12876, Japan
Takeharu
Shiraga
Takeharu Shiraga
Chuo University, Tokyo, Japan
JSPS KAKENHI Grant Number 19K20214, Japan
10.4230/LIPIcs.ICALP.2020.97
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.
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.
D. Aldous and J. Fill. Reversible Markov chains and random walks on graphs. URL: http://statwww.berkeley.edu/pub/users/aldous/RWG/book.html.
http://statwww.berkeley.edu/pub/users/aldous/RWG/book.html
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.
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.
I. Benjamini, S.-O. Chan, R. O'Donnell, O. Tamuzc, and L.-Y. Tand. Convergence, unanimity and disagreement in majority dynamics on unimodular graphs and random graphs. Stochastic Processes and their Applications, 126(9):2719-2733, 2016.
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.
P. Berenbrink, G. Giakkoupis, Anne-Marie Kermarrec, and F. Mallmann-Trenn. Bounds on the voter model in dynamic networks. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP), 2016.
E. Berger. Dynamic monopolies of constant size. Journal of Combinatorial Theory Series B, 83(2):191-200, 2001.
R. Pastor-Satorras C. Castellano, M. A. Muñoz. The non-linear q-voter model. Physical Review E, 80, 2009.
A. Clementi, M. Ghaffari, L. Gualà, E. Natale, F. Pasquale, and G. Scornavacca. A tight analysis of the parallel undecided-state dynamics with two colors. In Proceedings of the 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS), 117(28):1-15, 2018.
A. Coja-Oghlan. On the laplacian eigenvalues of G_n,p. Combinatorics, Probability and Computing, 16(6):923-946, 2007.
Nicholas Cook, Larry Goldstein, and Tobias Johnson. Size biased couplings and the spectral gap for random regular graphs. The Annals of Probability, 46(1):72-125, 2018.
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.
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.
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.
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.
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.
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.
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.
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.
M. Fischer, N. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1(1):26-39, 1986.
A. Frieze and M. Karońsky. Introduction to random graphs. Campridge University Press, 2016.
B. Gärtner and A. N. Zehmakan. Majority model on random regular graphs. In Proceedings of the 13th Latin American Symposium on Theoretical Informatics (LATIN), pages 572-583, 2018.
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.
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.
Y. Hassin and D. Peleg. Distributed probabilistic polling and applications to proportionate agreement. Information and Computation, 171(2):248-268, 2001.
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.
D. A. Levin and Y. Peres. Markov chain and mixing times: second edition. The American Mathematical Society, 2017.
T. M. Liggett. Interacting particle systems. Springer-Verlag, 1985.
R. Montenegro and P. Tetali. Mathematical aspects of mixing times in Markov chains. NOW Publishers, 2006.
E. Mossel, J. Neeman, and O. Tamuz. Majority dynamics and aggregation of information in social networks. Autonomous Agents and Multiagent Systems, 28(3):408-429, 2014.
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.
D. Peleg. Size bounds for dynamic monopolies. Discrete Applied Mathematics, 86(2-3):263-273, 1998.
D. Peleg. Local majorities, coalitions and monopolies in graphs: a review. Theoretical Computer Science, 282(2):231-257, 2002.
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.
N. Shimizu and T. Shiraga. Phase transitions of best-of-two and best-of-three on stochastic block models. In Proceedings of the 33rd International Symposium on Distributed Computing (DISC), pages 32:1-32:17, 2019.
N. Shimizu and T. Shiraga. Quasi-majority functional voting on expander graphs. arXiv, 2020. URL: http://arxiv.org/abs/2002.07411.
http://arxiv.org/abs/2002.07411
K. Tikhomirov and P. Youssef. The spectral gap of dense random regular graphs. The Annals of Probability, 47(1):362-419, 2019.
A. N. Zehmakan. Opinion forming in Erdős-Rényi random graph and expanders. In Proceedings of the 29th International Symposium on Algorithms and Computation (ISAAC), pages 4:1-4:13, 2018.
Nobutaka Shimizu and Takeharu Shiraga
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode