Abstract
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.
Highlights
We aim at modeling propagation of opinion over networks of the social type
We describe the discrete dynamical systems which come from the space and time discretization of standard Partial Differential Equations
11 w + w + Aw = 0, τ ητ with A = I − K, where K is a the stochastic matrix associated to the leftward shift, i.e. it exactly corresponds to the cycle which we considered previously
Summary
We aim at modeling propagation of opinion over networks of the social type. As a typical example, consider a collection of voters before the second round of a runoff voting process. The evolution models which are obtained in this way present some similarities with diffusion processes over networks, and the equation which. The topic of propagation of information over a network has been widely studied, both from the modeling and statistical standpoints, and we refer to Section 10 at the end of this article for a brief bibliographical survey. We will mainly concentrate on linear evolution models: our work is closely related to matrix theory and Markov chains, though the language and questionings will differ. Through these notes, we will revisit some well known results, but we hope that the reader will appreciate the relevance of the application of these results to our problem. In order to facilitate the reading, we have gathered most bibliographical notes in a dedicated section (Section 10) at the end of these lecture notes
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