### Lyapunov exponents of a Gaussian opinion dynamics model

Probability Seminar

3rd February 2023, 3:30 pm – 4:30 pm

Fry Building, 2.04

N individuals each hold numerical opinions on d topics, summarized as a vector in d-dimensional Euclidean space R^d. Every day, each individual independently updates their opinion, replacing it with a (fixed) convex combination of their opinion from the previous day and a sample from a multivariate Gaussian distribution. The mean and covariance of this Gaussian are respectively the empirical mean of the previous day’s N opinions, and alpha times the empirical covariance of the previous day’s opinions. The parameters of the model are the weights in the convex combination and the covariance scaling constant alpha.

We analyse the long-term behaviour of the opinions using the classical theory of Lyapunov exponents for iid random matrix products, and in particular a result of Newman (1986). We determine explicitly a critical value of alpha such that for alpha below the critical value, the opinions almost surely converge to a random limit, and for alpha at or above the critical value, the opinions almost surely diverge. Moreover, the d opinions almost surely become perfectly correlated asymptotically.

We end with a continuous-time analogue of the model, which is closely related to the right-invariant Brownian motion on GL(N,R) studied by Dynkin (1961), Orihara (1970) and Norris, Rogers and Williams, Baxendale and Harris, and Le Jan (all in 1986).

Joint work with Stas Volkov (Lund)

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