Issue |
ESAIM: ProcS
Volume 67, 2020
CEMRACS 2018 - Numerical and mathematical modeling for biological and medical applications: deterministic, probabilistic and statistical descriptions
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---|---|---|
Page(s) | 285 - 335 | |
DOI | https://doi.org/10.1051/proc/202067016 | |
Published online | 09 June 2020 |
Opinion propagation on social networks: a mathematical standpoint∗
1 Department of Mathematics, University of British Columbia, Vancouver BC Canada
2 DMA, École normale supérieure, CNRS, PSL University, 75005 Paris, France
& Laboratoire de Mathématiques d’Orsay, Univ. Paris Sud, CNRS, Univ. Paris-Saclay, Bâtiment 307, 91405 Orsay Cedex
These lecture notes address mathematical issues related to the modeling of opinion propagation on networks of the social type. Starting from the behavior of the simplest discrete linear model, we develop various standpoints and describe some extensions: stochastic interpretation, monitoring of a network, time continuous evolution problem, charismatic networks, links with discretized Partial Differential Equations, nonlinear models, inertial version and stability issues. These developments rely on basic mathematical tools, which makes them accessible at an undergraduate level. In a last section, we propose a new model of opinion propagation, where the opinion of an agent is described by a Gaussian density, and the (discrete) evolution equation is based on barycenters with respect to the Fisher metric.
Résumé
Ce support de cours traite de questions mathématiques en lien avec la modélisation de la propagation d’opinion dans des réseaux sociaux. À partir du modèle le plus simple, discret et linéaire, nous développons des points de vue divers et proposons des extensions: interprétation stochastique, contrôle des opinions sur un réseau, modèle d’évolution continu en temps, réseaux «charismatiques », liens avec les équations aux dérivées partielles discrétisées, modèles non linéaires, modèle avec inertie et questions de stabilité. Ces développements reposent sur un bagage limité d’outils mathématiques, de telle sorte que l’essentiel est accessible au niveau licence. Dans une dernière section, nous proposons un nouveau modèle de propagation d’opinion, où l’opinion est décrite par une densité gaussienne, et l’évolution discrète est basée sur la notion de barycentre selon la métrique de Fisher.
These notes follow a series of lectures which were given at two summerschools: Cemracs 2018, in Marseille, and Mathematical Summer in Paris, july 2018, at Ecole Normale Supérieure de Paris, PSL University.
The supplementary material (the final implementation) is available at https://www.esaim-proc.org/10.1051/proc/202067016/olm
© EDP Sciences, SMAI 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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