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On the steady state of continuous-time stochastic opinion dynamics with power-law confidence

Part of: Miscellaneous applications of operator theory

Published online by Cambridge University Press:  16 September 2021

François Baccelli Open the ORCID record for François Baccelli [Opens in a new window] ,
Sriram Vishwanath  and
Jae Oh Woo Open the ORCID record for Jae Oh Woo [Opens in a new window]
François Baccelli*
Affiliation:
The University of Texas at Austin and INRIA Paris
Sriram Vishwanath*
Affiliation:
The University of Texas at Austin
Jae Oh Woo*
Affiliation:
The University of Texas at Austin
*
*Postal address: Department of Mathematics, The University of Texas at Austin, 2515 Speedway, Austin, TX 78712, USA.
***Postal address: Department of Electrical and Computer Engineering, The University of Texas at Austin, 2501 Speedway, Austin, TX 78712, USA.
*Postal address: Department of Mathematics, The University of Texas at Austin, 2515 Speedway, Austin, TX 78712, USA.
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Abstract

This paper introduces a non-linear and continuous-time opinion dynamics model with additive noise and state-dependent interaction rates between agents. The model features interaction rates which are proportional to a negative power of the opinion distances. We establish a non-local partial differential equation for the distribution of opinion distances and use Mellin transforms to provide an explicit formula for the stationary solution of the latter, when it exists. Our approach leads to new qualitative and quantitative results on this type of dynamics. To the best of our knowledge these Mellin transform results are the first quantitative results on the equilibria of opinion dynamics with distance-dependent interaction rates. The closed-form expressions for this class of dynamics are obtained for the two-agent case. However, the results can be used in mean-field models featuring several agents whose interaction rates depend on the empirical average of their opinions. The technique also applies to linear dynamics, namely with a constant interaction rate, on an interaction graph.

Keywords

Opinion dynamicscontinuous-time Markov processMellin transformbounded confidencestochastic intensityadditive noisenon-local PDE

MSC classification

Primary: 47N30: Applications in probability theory and statistics
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 3 , September 2021 , pp. 746 - 772
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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