Macroscopic Noisy Bounded Confidence Models with Distributed Radical Opinions

In this article, we study the nonlinear Fokker-Planck (FP) equation that arises as a mean-field (macroscopic) approximation of bounded confidence opinion dynamics, where opinions are influenced by environmental noises and opinions of radicals (stubborn individuals). The distribution of radical opinions serves as an infinite-dimensional exogenous input to the FP equation, visibly influencing the steady opinion profile. We establish mathematical properties of the FP equation. In particular, we (i) show the well-posedness of the dynamic equation, (ii) provide existence result accompanied by a quantitative global estimate for the corresponding stationary solution, and (iii) establish an explicit lower bound on the noise level that guarantees exponential convergence of the dynamics to stationary state. Combining the results in (ii) and (iii) readily yields the input-output stability of the system for sufficiently large noises. Next, using Fourier analysis, the structure of opinion clusters under the uniform initial distribution is examined. Specifically, two numerical schemes for identification of order-disorder transition and characterization of initial clustering behavior are provided. The results of analysis are validated through several numerical simulations of the continuum-agent model (partial differential equation) and the corresponding discrete-agent model (interacting stochastic differential equations) for a particular distribution of radicals

Feedback Loops in Opinion Dynamics of Agent-Based Models with Multiplicative Noise

We introduce an agent-based model for co-evolving opinion and social dynamics, under the influence of multiplicative noise. In this model, every agent is characterized by a position in a social space and a continuous opinion state variable. Agents' movements are governed by positions and opinions of other agents and similarly, the opinion dynamics is influenced by agents' spatial proximity and their opinion similarity. Using numerical simulations and formal analysis, we study this feedback loop between opinion dynamics and mobility of agents in a social space. We investigate the behavior of this ABM in different regimes and explore the influence of various factors on appearance of emerging phenomena such as group formation and opinion consensus. We study the empirical distribution and in the limit of infinite number of agents we derive a corresponding reduced model given by a partial differential equation (PDE). Finally, using numerical examples we show that a resulting PDE model is a good approximation of the original ABM

Noisy bounded confidence models for opinion dynamics: the effect of boundary conditions on phase transitions

We study SDE and PDE models for opinion dynamics under bounded confidence, for a range of different boundary conditions, with and without the inclusion of a radical population. We perform exhaustive numerical studies with pseudo-spectral methods to determine the effects of the boundary conditions, suggesting that the no-flux case most faithfully reproduces the underlying mechanisms in the associated deterministic models of Hegselmann and Krause. We also compare the SDE and PDE models, and use tools from analysis to study phase transitions, including a systematic description of an appropriate order parameter

Dynamical Networks of Social Influence: Modern Trends and Perspectives

Dynamics and control of processes over social networks, such as the evolution of opinions, social influence and interpersonal appraisals, diffusion of information and misinformation, emergence and dissociation of communities, are now attracting significant attention from the broad research community that works on systems, control, identification and learning. To provide an introduction to this rapidly developing area, a Tutorial Session was included into the program of IFAC World Congress 2020. This paper provides a brief summary of the three tutorial lectures, covering the most “mature” directions in analysis of social networks and dynamics over them: 1) formation of opinions under social influence; 2) identification and learning for analysis of a network’s structure; 3) dynamics of interpersonal appraisals

Dynamical networks of social influence: Modern trends and perspectives

Dynamics and control of processes over social networks, such as the evolution of opinions, social influence and interpersonal appraisals, diffusion of information and misinfor-mation, emergence and dissociation of communities, are now attracting significant attention from the broad research community that works on systems, control, identification and learning. To provide an introduction to this rapidly developing area, a Tutorial Session was included into the program of IFAC World Congress 2020. This paper provides a brief summary of the three tutorial lectures, covering the most "mature"directions in analysis of social networks and dynamics over them: 1) formation of opinions under social influence; 2) identification and learning for analysis of a network's structure; 3) dynamics of interpersonal appraisals

Validating argument-based opinion dynamics with survey experiments

The empirical validation of models remains one of the most important challenges in opinion dynamics. In this contribution, we report on recent developments on combining data from survey experiments with computational models of opinion formation. We extend previous work on the empirical assessment of an argument-based model for opinion dynamics in which biased processing is the principle mechanism. While previous work (Banisch & Shamon, in press) has focused on calibrating the micro mechanism with experimental data on argument-induced opinion change, this paper concentrates on the macro level using the empirical data gathered in the survey experiment. For this purpose, the argument model is extended by an external source of balanced information which allows to control for the impact of peer influence processes relative to other noisy processes. We show that surveyed opinion distributions are matched with a high level of accuracy in a specific region in the parameter space, indicating an equal impact of social influence and external noise. More importantly, the estimated strength of biased processing given the macro data is compatible with those values that achieve high likelihood at the micro level. The main contribution of the paper is hence to show that the extended argument-based model provides a solid bridge from the micro processes of argument-induced attitude change to macro level opinion distributions. Beyond that, we review the development of argument-based models and present a new method for the automated classification of model outcomes.Comment: Keywords: opinion dynamics, validation, empirical confirmation, survey experiments, parameter estimation, argument communication theory, computational social scienc

A numerical framework for solving PDE-constrained optimization problems from multiscale particle dynamics

In this thesis, we develop accurate and efficient numerical methods for solving partial differential equation (PDE) constrained optimization problems arising from multiscale particle dynamics, with the aim of producing a desired time-dependent state at the minimal cost. A PDE-constrained optimization problem seeks to move one or more state variables towards a desired state under the influence of one or more control variables, and a set of constraints that are described by PDEs governing the behaviour of the variables. In particular, we consider problems constrained by one-dimensional and two-dimensional advection-diffusion problems with a non-local integral term, such as the associated mean-field limit Fokker-Planck equation of the noisy Hegselmann-Krause opinion dynamics model. We include additional bound constraints on the control variable for the opinion dynamics problem. Lastly, we consider constraints described by a two-dimensional robot swarming model made up of a system of advection-diffusion equations with additional linear and integral terms. We derive continuous Lagrangian first-order optimality conditions for these problems and solve the resulting systems numerically for the optimized state and control variables. Each of these problems, combined with Dirichlet, no-flux, or periodic boundary conditions, present unique challenges that require versatility of the numerical methods devised. Our numerical framework is based on a novel combination of four main components: (i) a discretization scheme, in both space and time, with the choice of pseudospectral or fi nite difference methods; (ii) a forward problem solver that is implemented via a differential-algebraic equation solver; (iii) an optimization problem solver that is a choice between a fi xed-point solver, with or without Armijo-Wolfe line search conditions, a Newton-Krylov algorithm, or a multiple shooting scheme, and; (iv) a primal-dual active set strategy to tackle additional bound constraints on the control variable. Pseudospectral methods efficiently produce highly accurate solutions by exploiting smoothness in the solutions, and are designed to perform very well with dense, small matrix systems. For a number of problems, we take advantage of the exponential convergence of pseudospectral methods by discretising in this way not only in space, but also in time. The alternative fi nite difference method performs comparatively well when non-smooth bound constraints are added to the optimization problem. A differential{algebraic equation solver works out the discretized PDE on the interior of the domain, and applies the boundary conditions as algebraic equations. This ensures generalizability of the numerical method, as one does not need to explicitly adapt the numerical method for different boundary conditions, only to specify different algebraic constraints that correspond to the boundary conditions. A general fixed-point or sweeping method solves the system of equations iteratively, and does not require the analytic computation of the Jacobian. We improve the computational speed of the fi xed-point solver by including an adaptive Armijo-Wolfe type line search algorithm for fixed-point problems. This combination is applicable to problems with additional bound constraints as well as to other systems for which the regularity of the solution is not sufficient to be exploited by the spectral-in-space-and-time nature of the Newton-Krylov approach. The recently devised Newton-Krylov scheme is a higher-order, more efficient optimization solver which efficiently describes the PDEs and the associated Jacobian on the discrete level, as well as solving the resulting Newton system efficiently via a bespoke preconditioner. However, it requires the computation of the Jacobian, and could potentially be more challenging to adapt to more general problems. Multiple shooting solves an initial-value problem on sections of the time interval and imposes matching conditions to form a solution on the whole interval. The primal-dual active set strategy is used for solving our non-linear and non-local optimization problems obtained from opinion dynamics problems, with pointwise non-equality constraints. This thesis provides a numerical framework that is versatile and generalizable for solving complex PDE-constrained optimization problems from multiscale particle dynamic

Kinetic Theory and Swarming Tools to Modeling Complex Systems—Symmetry problems in the Science of Living Systems

This MPDI book comprises a number of selected contributions to a Special Issue devoted to the modeling and simulation of living systems based on developments in kinetic mathematical tools. The focus is on a fascinating research field which cannot be tackled by the approach of the so-called hard sciences—specifically mathematics—without the invention of new methods in view of a new mathematical theory. The contents proposed by eight contributions witness the growing interest of scientists this field. The first contribution is an editorial paper which presents the motivations for studying the mathematics and physics of living systems within the framework an interdisciplinary approach, where mathematics and physics interact with specific fields of the class of systems object of modeling and simulations. The different contributions refer to economy, collective learning, cell motion, vehicular traffic, crowd dynamics, and social swarms. The key problem towards modeling consists in capturing the complexity features of living systems. All articles refer to large systems of interaction living entities and follow, towards modeling, a common rationale which consists firstly in representing the system by a probability distribution over the microscopic state of the said entities, secondly, in deriving a general mathematical structure deemed to provide the conceptual basis for the derivation of models and, finally, in implementing the said structure by models of interactions at the microscopic scale. Therefore, the modeling approach transfers the dynamics at the low scale to collective behaviors. Interactions are modeled by theoretical tools of stochastic game theory. Overall, the interested reader will find, in the contents, a forward look comprising various research perspectives and issues, followed by hints on to tackle these