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Distribution of consensus in a broadcast-based consensus algorithm with random initial opinions

Part of: Mathematical sociology Limit theorems Distribution theory - Probability

Published online by Cambridge University Press:  09 May 2023

Shigeo Shioda  and
Dai Kato
Shigeo Shioda*
Affiliation:
Chiba University
Dai Kato*
Affiliation:
Chiba University
*
*Postal address: Graduate School of Engineering, Chiba University, 1-33, Yayoi, Inage, Chiba, 263-8522 Japan.
*Postal address: Graduate School of Engineering, Chiba University, 1-33, Yayoi, Inage, Chiba, 263-8522 Japan.
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Abstract

We study the distribution of the consensus formed by a broadcast-based consensus algorithm for cases in which the initial opinions of agents are random variables. We first derive two fundamental equations for the time evolution of the average opinion of agents. Using the derived equations, we then investigate the distribution of the consensus in the limit in which agents do not have any mutual trust, and show that the consensus without mutual trust among agents is in sharp contrast to the consensus with complete mutual trust in the statistical properties if the initial opinion of each agent is integrable. Next, we provide the formulation necessary to mathematically discuss the consensus in the limit in which the number of agents tends to infinity, and derive several results, including a central limit theorem concerning the consensus in this limit. Finally, we study the distribution of the consensus when the initial opinions of agents follow a stable distribution, and show that the consensus also follows a stable distribution in the limit in which the number of agents tends to infinity.

Keywords

Stable distributioncentral limit theoremCauchyGaussianLévy

MSC classification

Primary: 91D30: Social networks
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60F05: Central limit and other weak theorems
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 4 , December 2023 , pp. 1416 - 1438
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Acemoğlu, D., Como, G., Fagnani, F. and Ozdaglar, A. (2011). Opinion fluctuations and persistent disagreement in social networks. In 2011 50th IEEE Conference on Decision and Control and European Control Conference, pp. 2347–2352. IEEE.CrossRefGoogle Scholar
Acemoğlu, D., Como, G., Fagnani, F. and Ozdaglar, A. (2013). Opinion fluctuations and disagreement in social networks. Math. Operat. Res. 38, 1–27.Google Scholar
Altafini, C. and Lini, G. (2014). Predictable dynamics of opinion forming for networks with antagonistic interactions. IEEE Trans. Automatic Control 60, 342–357.Google Scholar
Aysal, T. C., Yildiz, M. E., Sarwate, A. D. and Scaglione, A. (2009). Broadcast gossip algorithms for consensus. IEEE Trans. Signal Process. 57, 2748–2761.Google Scholar
DeGroot, M. H. (1974). Reaching a consensus. J. Amer. Statist. Assoc. 69, 118–121.Google Scholar
Durrett, R. (2019). Probability: Theory and Examples, 5th edn. Cambridge University Press.Google Scholar
Fagnani, F. and Zampieri, S. (2008). Randomized consensus algorithms over large scale networks. IEEE J. Sel. Areas Commun. 26, 634–649.CrossRefGoogle Scholar
Frasca, P., Ravazzi, C., Tempo, R. and Ishii, H. (2013). Gossips and prejudices: ergodic randomized dynamics in social networks. IFAC Proceedings Volumes 46, 212–219.CrossRefGoogle Scholar
Katoh, D. and Shioda, S. (2022). Probability laws of consensus in a broadcast-based consensus-forming algorithm. Stoch. Models 38, 91–115.Google Scholar
Li, Q. and Rus, D. (2006). Global clock synchronization in sensor networks. IEEE Trans. Comput. 55, 214–226.Google Scholar
Muthukrishnan, S., Ghosh, B. and Schultz, M. H. (1998). First- and second-order diffusive methods for rapid, coarse, distributed load balancing. Theory Comput. Syst. 31, 331–354.CrossRefGoogle Scholar
Nolan, J. P. (2020). Stable Distributions: Models for Heavy Tailed Data. Springer.CrossRefGoogle Scholar
Olfati-Saber, R. and Murray, R. M. (2004). Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Automatic Control 49, 1520–1533.CrossRefGoogle Scholar
Rabani, Y., Sinclair, A. and Wanka, R. (1998). Local divergence of Markov chains and the analysis of iterative load-balancing schemes. In 39th Annual Symposium on Foundations of Computer Science, pp. 694–703. IEEE.Google Scholar
Shioda, S. (2020). Distribution of consensus in a broadcasting-based consensus-forming algorithm. ACM SIGMETRICS Performance Evaluation Rev. 48, 91–96.Google Scholar
Shioda, S. and Takehara, K. (2022). Ergodicity of time reversal process of stochastic consensus formation and its application. In ACM SIGMETRICS Performance Evaluation Rev. 50, 12–14.Google Scholar
Silvestre, D., Hespanha, J. P. and Silvestre, C. (2019). Broadcast and gossip stochastic average consensus algorithms in directed topologies. IEEE Trans. Control Network Syst. 6, 474–486.CrossRefGoogle Scholar
Tsitsiklis, J. N. (1984). Problems in decentralized decision making and computation. Tech. Rep. Massachusetts Institute of Technology Cambridge Laboratory for Information and Decision Systems.Google Scholar
Xiao, L., Boyd, S. and Lall, S. (2005). A scheme for robust distributed sensor fusion based on average consensus. In Fourth International Symposium on Information Processing in Sensor Networks (IPSN 2005), pp. 63–70. IEEE.Google Scholar
Yang, Y. and Blum, R. S. (2013). Broadcast-based consensus with non-zero-mean stochastic perturbations. IEEE Trans. Inform. Theory 59, 3971–3989.Google Scholar