14 research outputs found
Opinion propagation on social networks: a mathematical standpoint∗
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
Functional Maps Representation on Product Manifolds
We consider the tasks of representing, analyzing and manipulating maps
between shapes. We model maps as densities over the product manifold of the
input shapes; these densities can be treated as scalar functions and therefore
are manipulable using the language of signal processing on manifolds. Being a
manifold itself, the product space endows the set of maps with a geometry of
its own, which we exploit to define map operations in the spectral domain; we
also derive relationships with other existing representations (soft maps and
functional maps). To apply these ideas in practice, we discretize product
manifolds and their Laplace--Beltrami operators, and we introduce localized
spectral analysis of the product manifold as a novel tool for map processing.
Our framework applies to maps defined between and across 2D and 3D shapes
without requiring special adjustment, and it can be implemented efficiently
with simple operations on sparse matrices.Comment: Accepted to Computer Graphics Foru
Opinion propagation on social networks: a mathematical standpoint
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