Solution of the multi-state voter model and application to strong neutrals in the naming game

Abstract

We consider the voter model with $M$ states initially in the system. Using generating functions, we pose the spectral problem for the Markov transition matrix and solve for all eigenvalues and eigenvectors exactly. With this solution, we can find all future probability probability distributions, the expected time for the system to condense from $M$ states to $M-1$ states, the moments of consensus time, the expected local times, and the expected number of states over time. Furthermore, when the initial distribution is uniform, such as when $M=N$, we can find simplified expressions for these quantities. In particular, we show that the mean and variance of consensus time for $M=N$ is $\frac{1}{N}(N-1{)}^2$ and $\frac{1}{3}({\pi }^2-9)(N-1{)}^2$ respectively.

10 pages, 4 figures, 2 appendices

References in corpus (12)

• Statistical physics of social dynamics
• Sharp transition towards shared vocabularies in multi-agent systems
• Does a Single Zealot Affect an Infinite Group of Voters ?
• On the Role of Zealotry in the Voter Model
• Stochastic Models of Evolution in Genetics, Ecology and Linguistics
• Ordering dynamics with two non-excluding options: Bilingualism in language competition
• Non-equilibrium phase transition in negotiation dynamics
• The Naming Game in Social Networks: Community Formation and Consensus Engineering
• In-depth analysis of the Naming Game dynamics: the homogeneous mixing case
• Social Influencing and Associated Random Walk Models: Asymptotic Consensus Times on the Complete Graph
• Dynamics of Strategic Three-Choice Voting
• Solution of the Voter Model by Spectral Analysis

Cited by in corpus (11)

• A multi-state voter model with imperfect copying
• Zealots in multi-state noisy voter models
• Analysis of the high dimensional naming game with committed minorities
• Filter Bubble effect in the multistate voter model
• Local and global ordering dynamics in multi-state voter models
• Polyadic Opinion Formation: The Adaptive Voter Model on a Hypergraph
• Finite-size effects on the convergence time in continuous-opinion dynamics
• Flocking dynamics with voter-like interactions
• Approach to consensus in models of continuous-opinion dynamics: a study inspired by the physics of granular gases
• Analysis of a voter model with an evolving number of opinion states
• Noisy multistate voter model for flocking in finite dimensions