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Asymptotic and Stability Analysis of Kinetic Models for Opinion Formation on Networks: An Allen--Cahn Approach.

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Title: Asymptotic and Stability Analysis of Kinetic Models for Opinion Formation on Networks: An Allen--Cahn Approach.
Authors: Burger, Martin1 martin.burger@desy.de, Loy, Nadia2 nadia.loy@polito.it, Rossi, Alex3 alex.rossi@fau.de
Source: SIAM Journal on Applied Dynamical Systems. 2025, Vol. 24 Issue 2, p1042-1069. 28p.
Subjects: Boltzmann's equation, Integral operators, Social norms, Linear statistical models, Linear equations
Abstract: We present the analysis of stationary equilibria and their stability in the case of an opinion formation process in the presence of binary opposite opinions evolving according to majority-like rules on social networks. The starting point is a kinetic Boltzmann-type model implementing microscopic interaction rules that can be either binary or ternary for the opinion exchange among individuals holding a certain degree of connectivity. The key idea is to derive from the kinetic model an Allen--Cahn type equation for the fraction of individuals holding one of the two opinions. The latter can be studied by means of a linear stability analysis and by exploiting integral operator analysis. While this is true for ternary interactions, for binary interactions the derived equation of interest is a linear scattering equation that can be studied by means of general relative entropy tools and integral operators. We extend the analysis to a continuous opinion model and coevolving networks. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Applied Dynamical Systems is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22SIAM+Journal+on+Applied+Dynamical+Systems%22">SIAM Journal on Applied Dynamical Systems</searchLink>. 2025, Vol. 24 Issue 2, p1042-1069. 28p.
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  Data: We present the analysis of stationary equilibria and their stability in the case of an opinion formation process in the presence of binary opposite opinions evolving according to majority-like rules on social networks. The starting point is a kinetic Boltzmann-type model implementing microscopic interaction rules that can be either binary or ternary for the opinion exchange among individuals holding a certain degree of connectivity. The key idea is to derive from the kinetic model an Allen--Cahn type equation for the fraction of individuals holding one of the two opinions. The latter can be studied by means of a linear stability analysis and by exploiting integral operator analysis. While this is true for ternary interactions, for binary interactions the derived equation of interest is a linear scattering equation that can be studied by means of general relative entropy tools and integral operators. We extend the analysis to a continuous opinion model and coevolving networks. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of SIAM Journal on Applied Dynamical Systems is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1137/24M1671128
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        Text: English
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      – SubjectFull: Boltzmann's equation
        Type: general
      – SubjectFull: Integral operators
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      – SubjectFull: Social norms
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      – SubjectFull: Linear statistical models
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      – SubjectFull: Linear equations
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      – TitleFull: Asymptotic and Stability Analysis of Kinetic Models for Opinion Formation on Networks: An Allen--Cahn Approach.
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              Text: 2025
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