Approach to consensus in models of continuous-opinion dynamics: a study inspired by the physics of granular gases

Abstract

A model for continuous-opinion dynamics is proposed and studied by taking advantage of its similarities with a mono-dimensional granular gas. Agents interact as in the Deffuant model, with a parameter $\alpha$ controlling the persuasibility of the individuals. The interaction coincides with the collision rule of two grains moving on a line, provided opinions and velocities are identified, with $\alpha$ being the so-called coefficient of normal restitution. Starting from the master equation of the probability density of all opinions, general conditions are given for the system to reach consensus. The case when the interaction frequency is proportional to the $\beta$-power of the relative opinions is studied in more detail. It is shown that the mean-field approximation to the master equation leads to the Boltzmann kinetic equation for the opinion distribution. In this case, the system always approaches consensus, which can be seen as the approach to zero of the opinion temperature, a measure of the width of the opinion distribution. Moreover, the long-time behaviour of the system is characterized by a scaling solution to the Boltzmann equation in which all time dependence occurs through the temperature. The case $\beta =0$ is related to the Deffuant model and is analytically soluble. The scaling distribution is unimodal and independent of $\alpha$. For $\beta >0$ the distribution of opinions is unimodal below a critical value of $|\alpha |$, being multimodal with two maxima above it. This means that agents may approach consensus while being polarized. Near the critical points and for $|\alpha |\ge 0.4$, the distribution of opinions is well approximated by the sum of two Gaussian distributions. Monte Carlo simulations are in agreement with the theoretical results.

arXiv comment18 pages, 3 figures

References in corpus (44)

• Statistical physics of social dynamics
• A Tutorial on Modeling and Analysis of Dynamic Social Networks. Part I
• Continuous Opinion Dynamics under Bounded Confidence: A Survey
• Homophily, Cultural Drift and the Co-Evolution of Cultural Groups
• A stabilization theorem for dynamics of continuous opinions
• A Tutorial on Modeling and Analysis of Dynamic Social Networks. Part II
• Voter model dynamics in complex networks: Role of dimensionality, disorder and degree distribution
• Bifurcations and Patterns in Compromise Processes
• Multiscaling in Infinite Dimensional Collision Processes
• Coarsening and persistence in the voter model
• Freezing and Slow Evolution in a Constrained Opinion Dynamics Model
• Ordering dynamics with two non-excluding options: Bilingualism in language competition
• Opinion dynamics: rise and fall of political parties
• Decelerating microdynamics can accelerate macrodynamics in the voter model
• Influence of correlations on the velocity statistics of scalar granular gases
• Zealots in the mean-field noisy voter model
• Mass media and repulsive interactions in continuous-opinion dynamics
• Noisy continuous--opinion dynamics
• Update rules and interevent time distributions: Slow ordering vs. no ordering in the Voter Model
• A model for the atomic-scale structure of a dense, nonequilibrium fluid: the homogeneous cooling state of granular fluids
• The role of noise and initial conditions in the asymptotic solution of a bounded confidence, continuous-opinion model
• Consensus formation on coevolving networks: groups' formation and structure
• Exact Solution of the Multi-Allelic Diffusion Model
• Ordering dynamics of the multi-state voter model
• Diffusing opinions in bounded confidence processes
• The noisy voter model under the influence of contrarians
• Noisy bounded confidence models for opinion dynamics: the effect of boundary conditions on phase transitions
• Spatial correlations in vote statistics: a diffusive field model for decision-making
• Consensus and diversity in multi-state noisy voter models
• Ordering dynamics in the voter model with aging
• First-passage distributions for the one-dimensional Fokker-Planck equation
• Algebraic coarsening in voter models with intermediate states
• Aging-induced continuous phase transition
• Steady state representation of the homogeneous cooling state of a granular gas
• Zealots in multi-state noisy voter models
• On the emergence of an "intention field" for socially cohesive agents
• Solution of the multi-state voter model and application to strong neutrals in the naming game
• Reduction from non-Markovian to Markovian dynamics: The case of aging in the noisy-voter model
• The Boltzmann legacy revisited: kinetic models of social interactions
• Asymptotic solutions of the nonlinear Boltzmann equation for dissipative systems
• Markov jump processes and collision-like models in the kinetic description of multi-agent systems
• Generalized time evolution of the homogeneous cooling state of a granular gas with positive and negative coefficient of normal restitution
• Heat flux of a granular gas with homogeneous temperature
• Velocity fluctuations in cooling granular gases

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