[Paper] Consensus Time in 3-Majority and 2-Choices Is Determined by the Maximum Initial Opinion Density
Source: arXiv - 2606.11778v1
Overview
We establish the correct parameter governing the convergence time of the 3-Majority and 2-Choices dynamics on the complete graph in the synchronous model. Recent work [Shimizu and Shiraga, PODC’25] provides matching upper and lower bounds on the number of rounds to consensus, but only in a weak sense: the bounds are shown to coincide for some initial opinion configuration. In contrast, we obtain tight bounds in a strong sense, with upper and lower bounds matching up to logarithmic factors for every initial configuration. Let $α$ (0) be the initial opinion-frequency vector, and denote by $α$ (0) ___ $\infty$ its maximum entry. We show that 3-Majority reaches consensus in $Θ$(min{$α$ (0) ___ -1 $\infty$ , $\sqrt$ n}) rounds w.h.p., while 2-Choices reaches consensus in $Θ$(___$α$ (0) ___ -1 $\infty$ ) rounds w.h.p. Our results demonstrate that the convergence time of both dynamics is governed not by global parameters such as the number of opinions k or the squared ${\ell}$ 2 norm of the initial opinion distribution, but rather by the ”local” parameter ___$α$ (0) ___ $\infty$ , the maximum initial opinion density.
Key Contributions
This paper presents research in the following areas:
- cs.DC
Methodology
Please refer to the full paper for detailed methodology.
Practical Implications
This research contributes to the advancement of cs.DC.
Authors
- Niccolò D Archivio
Paper Information
- arXiv ID: 2606.11778v1
- Categories: cs.DC
- Published: June 10, 2026
- PDF: Download PDF