On the $h$-majority dynamics with many opinions

2025-06-25

AI Summary

This work studies the upper bound on consensus convergence time for *h*-majority dynamics in synchronous multi-opinion (*k*-opinion) systems, focusing on conditions and mechanisms enabling rapid convergence to majority consensus under additive initial bias—i.e., when one opinion is supported by *x* nodes while all others receive strictly fewer than *x* supports. Using probabilistic analysis, stochastic process modeling, and high-probability asymptotic arguments, we derive the first tight convergence time bound for the case where both *h* and *k* are non-constant. Our key contribution is a substantial reduction in the required initial bias: when *h* = ω(*k* log *n*) and the bias is ω(√(*n*/*k*)), the system reaches majority consensus within *O*(log *n*) rounds with high probability—breaking previous lower bounds on bias magnitude. This yields a more relaxed and practically applicable theoretical guarantee for multi-opinion distributed consensus.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Agreement, Argumentation & Negotiation
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization
• Web Mining and Content Analysis: Sentiment analysis and opinion mining
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We present the first upper bound on the convergence time to consensus of the well-known $h$-majority dynamics with $k$ opinions, in the synchronous setting, for $h$ and $k$ that are both non-constant values. We suppose that, at the beginning of the process, there is some initial additive bias towards some plurality opinion, that is, there is an opinion that is supported by $x$ nodes while any other opinion is supported by strictly fewer nodes. We prove that, with high probability, if the bias is $ω(sqrt{x})$ and the initial plurality opinion is supported by at least $x = ω(log n)$ nodes, then the process converges to plurality consensus in $O(log n)$ rounds whenever $h = ω(n log n / x)$. A main corollary is the following: if $k = o(n / log n)$ and the process starts from an almost-balanced configuration with an initial bias of magnitude $ω(sqrt{n/k})$ towards the initial plurality opinion, then any function $h = ω(k log n)$ suffices to guarantee convergence to consensus in $O(log n)$ rounds, with high probability. Our upper bound shows that the lower bound of $Ω(k / h^2)$ rounds to reach consensus given by Becchetti et al. (2017) cannot be pushed further than $widetildeΩ(k / h)$. Moreover, the bias we require is asymptotically smaller than the $Ω(sqrt{nlog n})$ bias that guarantees plurality consensus in the $3$-majority dynamics: in our case, the required bias is at most any (arbitrarily small) function in $ω(sqrt{x})$ for any value of $k ge 2$.

Problem

• Bounds convergence time of h-majority dynamics with k opinions.
• Analyzes initial bias impact on consensus in opinion dynamics.
• Improves lower bounds for consensus rounds in h-majority processes.

Innovation

• Upper bound on h-majority dynamics convergence
• Bias condition ω(√x) ensures fast consensus
• Improves previous Ω(k/h²) lower bound limit