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Convex Analysisの二,三の進展について

Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

Opinion dynamics have attracted the interest of researchers from different fields. Local interactions among individuals create interesting dynamics for the system as a whole. Such dynamics are important from a variety of perspectives. Group decision making, successful marketing, and constructing networks (in which consensus can be reached or prevented) are a few examples of existing or potential applications. The invention of the Internet has made the opinion fusion faster, unilateral, and on a whole different scale. Spread of fake news, propaganda, and election interferences have made it clear there is an essential need to know more about these dynamics. The emergence of new ideas in the field has accelerated over the last few years. In the first quarter of 2020, at least 50 research papers have emerged, either peer-reviewed and published or on preprint outlets such as arXiv. In this paper, we summarize these ground-breaking ideas and their fascinating extensions and introduce newly surfaced concepts.

Recent advances in opinion propagation dynamics: A 2020 Survey

Simulation and the Monte Carlo Method

We consider a system of urns of Polya-type, with balls of two colors; the reinforcement of each urn depends both on the content of the same urn and on the average content of all urns. We show that the urns synchronize almost surely, in the sense that the fraction of balls of a given color converges almost surely, as the time goes to infinity, to the same limit for all urns. A normal approximation for a large number of urns is also obtained.

Synchronization via interacting reinforcement

We consider a model of N interacting two-colour Friedman urns. The interaction model considered is such that the reinforcement of each urn depends on the fraction of balls of a particular colour in that urn as well as the overall fraction of balls of that colour in all the urns combined together. We show that the urns synchronize almost surely and that the fraction of balls of each colour converges to the deterministic limit of one-half, which matches with the limit known for a single Friedman urn. Furthermore, we use the notion of stable convergence to obtain limit theorems for fluctuations around the synchronization limit.

Synchronization and fluctuation theorems for interacting Friedman urns

The focus of this work is on designing influencing strategies to shape the collective opinion of a network of individuals. We consider a variant of the voter model where opinions evolve in one of two ways. In the absence of external influence, opinions evolve via interactions between individuals in the network, while, in the presence of external influence, opinions shift in the direction preferred by the influencer. We focus on a finite time-horizon and an influencing strategy is characterized by when it exerts influence in this time-horizon given its budget constraints. Prior work on this opinion dynamics model assumes that individuals take into account the opinion of all individuals in the network. We generalize this and consider the setting where the opinion evolution of an individual depends on a limited collection of opinions from the network. We characterize the nature of optimal influencing strategies as a function of the way in which this collection of opinions is formed.

Influencing Opinion Dynamics in Networks with Limited Interaction

We introduce a general two colour interacting urn model on a finite directed graph, where each urn at a node, reinforces all the urns in its out-neighbours according to a fixed, non-negative and balanced reinforcement matrix. We show that the fraction of balls of either colour converges almost surely to a deterministic limit if either the reinforcement is not of Polya type or if the graph is such that every vertex with non-zero in-degree can be reached from some vertex with zero in-degree. We also obtain joint central limit theorems, with appropriate scaling, around the vector of limiting proportion. Further, in the remaining case when there are no vertices with zero in-degree and the reinforcement is of Polya type, we restrict our analysis to a regular graph and show that the fraction of balls of either colour converges almost surely to a finite random limit, which is the same across all the urns.

Interacting Urns on a Finite Directed Graph.

Gaussian approximations in high dimensional estimation

This paper aims at achieving a "good" estimator for the gradient of a function on a high-dimensional space. Often such functions are not sensitive in all coordinates and the gradient of the function is almost sparse. We propose a method for gradient estimation that combines ideas from Spall's Simultaneous Perturbation Stochastic Approximation with compressive sensing. The aim is to obtain "good" estimator without too many function evaluations. Application to estimating gradient outer product matrix as well as standard optimization problems are illustrated via simulations.

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Neeraja Sahasrabudhe

Papers

Synchronization and fluctuation theorems for interacting Friedman urns

Influencing Opinion Dynamics in Networks with Limited Interaction

Interacting Urns on a Finite Directed Graph.

Gaussian approximations in high dimensional estimation

Gradient Estimation with Simultaneous Perturbation and Compressive Sensing