Gaussian consensus processes and their Lyapunov exponents
Crane, Edward and Volkov, Stanislav LU
(2026)
In Electronic Journal of Probability
31.
- Abstract
We introduce a simple dynamic model of opinion formation, in which a finite population of individuals hold vector-valued opinions. At each time step, each individual’s opinion moves towards the mean opinion but is then perturbed independently by a centred multivariate Gaussian random variable, with covariance proportional to the covariance matrix of the opinions of the population. We establish precise necessary and sufficient conditions on the parameters of the model, under which all opinions converge to a common limiting value. Asymptotically perfect correlation emerges between opinions on different topics. Our results are rigorous and based on properties of the partial products of an i.i.d. sequence of random matrices. Each matrix is... (More)
We introduce a simple dynamic model of opinion formation, in which a finite population of individuals hold vector-valued opinions. At each time step, each individual’s opinion moves towards the mean opinion but is then perturbed independently by a centred multivariate Gaussian random variable, with covariance proportional to the covariance matrix of the opinions of the population. We establish precise necessary and sufficient conditions on the parameters of the model, under which all opinions converge to a common limiting value. Asymptotically perfect correlation emerges between opinions on different topics. Our results are rigorous and based on properties of the partial products of an i.i.d. sequence of random matrices. Each matrix is a fixed linear combination of the identity matrix and a real Ginibre matrix. We derive an analytic expression for the maximal Lyapunov exponent of this product sequence. We also analyze a continuous-time analogue of our model.
(Less)
- author
- Crane, Edward
and Volkov, Stanislav
LU
- organization
- publishing date
- 2026
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- 60F15, 93D50, Brownian motion of ellipsoids. MSC2020 subject classifications: 60G20, consensus process, DeGroot learning, interacting particle systems, Lyapunov exponents, phase transition, real Ginibre ensemble, stochastic opinion dynamics
- in
- Electronic Journal of Probability
- volume
- 31
- article number
- 74
- publisher
- UNIV WASHINGTON, DEPT MATHEMATICS
- external identifiers
-
- scopus:105040017560
- ISSN
- 1083-6489
- DOI
- 10.1214/26-EJP1527
- language
- English
- LU publication?
- yes
- id
- 9eda04d3-3160-45f2-909e-29e5876649c5
- date added to LUP
- 2026-09-14 14:21:22
- date last changed
- 2026-09-14 14:21:59
@article{9eda04d3-3160-45f2-909e-29e5876649c5,
abstract = {{<p>We introduce a simple dynamic model of opinion formation, in which a finite population of individuals hold vector-valued opinions. At each time step, each individual’s opinion moves towards the mean opinion but is then perturbed independently by a centred multivariate Gaussian random variable, with covariance proportional to the covariance matrix of the opinions of the population. We establish precise necessary and sufficient conditions on the parameters of the model, under which all opinions converge to a common limiting value. Asymptotically perfect correlation emerges between opinions on different topics. Our results are rigorous and based on properties of the partial products of an i.i.d. sequence of random matrices. Each matrix is a fixed linear combination of the identity matrix and a real Ginibre matrix. We derive an analytic expression for the maximal Lyapunov exponent of this product sequence. We also analyze a continuous-time analogue of our model.</p>}},
author = {{Crane, Edward and Volkov, Stanislav}},
issn = {{1083-6489}},
keywords = {{60F15; 93D50; Brownian motion of ellipsoids. MSC2020 subject classifications: 60G20; consensus process; DeGroot learning; interacting particle systems; Lyapunov exponents; phase transition; real Ginibre ensemble; stochastic opinion dynamics}},
language = {{eng}},
publisher = {{UNIV WASHINGTON, DEPT MATHEMATICS}},
series = {{Electronic Journal of Probability}},
title = {{Gaussian consensus processes and their Lyapunov exponents}},
url = {{http://dx.doi.org/10.1214/26-EJP1527}},
doi = {{10.1214/26-EJP1527}},
volume = {{31}},
year = {{2026}},
}