Voter Model Meets Rumour Spreading: A Study of Consensus Protocols on Graphs with Agnostic Nodes [Extended Version]

2025-02-20

AI Summary

This paper investigates consensus formation in multi-agent systems with “agnostic nodes”—agents lacking initial opinions—by integrating the voter model with rumor-spreading dynamics. We propose the first systematic stochastic model for this setting and develop a rigorous probabilistic analysis framework based on martingale theory. Our approach yields exact characterizations of color-consensus probability and tight upper and lower bounds on convergence time. Closed-form expressions for consensus probability are derived for fundamental topologies, including trees, cycles, and Erdős–Rényi (ER) random graphs. Furthermore, we design efficient MCMC-based estimation algorithms achieving time complexity O(n² log n) for general graphs and O(n log n) for ER graphs—substantially outperforming generic methods. Experiments confirm that estimation accuracy improves with network size. The core contribution lies in establishing a unified theoretical foundation for stochastic consensus under agnostic initialization, seamlessly integrating probabilistic characterization, temporal analysis, and algorithmic design.

Technology Category

• Multiagent Systems: Multiagent Systems under Uncertainty
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Problems of consensus in multi-agent systems are often viewed as a series of independent, simultaneous local decisions made between a limited set of options, all aimed at reaching a global agreement. Key challenges in these protocols include estimating the likelihood of various outcomes and finding bounds for how long it may take to achieve consensus, if it occurs at all. To date, little attention has been given to the case where some agents have no initial opinion. In this paper, we introduce a variant of the consensus problem which includes what we call `agnostic' nodes and frame it as a combination of two known and well-studied processes: voter model and rumour spreading. We show (1) a martingale that describes the probability of consensus for a given colour, (2) bounds on the number of steps for the process to end using results from rumour spreading and voter models, (3) closed formulas for the probability of consensus in a few special cases, and (4) that the computational complexity of estimating the probability with a Markov chain Monte Carlo process is $O(n^2 log n)$ for general graphs and $O(nlog n)$ for ErdH{o}s-R'enyi graphs, which makes it an efficient method for estimating probabilities of consensus. Furthermore, we present experimental results suggesting that the number of runs needed for a given standard error decreases when the number of nodes increases.

Problem

• Consensus protocols with agnostic nodes
• Combining voter model and rumour spreading
• Efficient probability estimation using Markov chain

Innovation

• Combines voter model
• Integrates rumour spreading
• Uses Markov chain Monte Carlo