153 research outputs found
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
On a Modified DeGroot-Friedkin Model of Opinion Dynamics
This paper studies the opinion dynamics that result when individuals
consecutively discuss a sequence of issues. Specifically, we study how
individuals' self-confidence levels evolve via a reflected appraisal mechanism.
Motivated by the DeGroot-Friedkin model, we propose a Modified DeGroot-Friedkin
model which allows individuals to update their self-confidence levels by only
interacting with their neighbors and in particular, the modified model allows
the update of self-confidence levels to take place in finite time without
waiting for the opinion process to reach a consensus on any particular issue.
We study properties of this Modified DeGroot-Friedkin model and compare the
associated equilibria and stability with those of the original DeGroot-Friedkin
model. Specifically, for the case when the interaction matrix is doubly
stochastic, we show that for the modified model, the vector of individuals'
self-confidence levels asymptotically converges to a unique nontrivial
equilibrium which for each individual is equal to 1/n, where n is the number of
individuals. This implies that eventually, individuals reach a democratic
state
Consensus Convergence with Stochastic Effects
We consider a stochastic, continuous state and time opinion model where each
agent's opinion locally interacts with other agents' opinions in the system,
and there is also exogenous randomness. The interaction tends to create
clusters of common opinion. By using linear stability analysis of the
associated nonlinear Fokker-Planck equation that governs the empirical density
of opinions in the limit of infinitely many agents, we can estimate the number
of clusters, the time to cluster formation and the critical strength of
randomness so as to have cluster formation. We also discuss the cluster
dynamics after their formation, the width and the effective diffusivity of the
clusters. Finally, the long term behavior of clusters is explored numerically.
Extensive numerical simulations confirm our analytical findings.Comment: Dedication to Willi J\"{a}ger's 75th Birthda
Optimal control of the convergence time in the Hegselmann--Krause dynamics
We study the optimal control problem of minimizing the convergence time in
the discrete Hegselmann--Krause model of opinion dynamics. The underlying model
is extended with a set of strategic agents that can freely place their opinion
at every time step. Indeed, if suitably coordinated, the strategic agents can
significantly lower the convergence time of an instance of the
Hegselmann--Krause model. We give several lower and upper worst-case bounds for
the convergence time of a Hegselmann--Krause system with a given number of
strategic agents, while still leaving some gaps for future research.Comment: 14 page
On symmetric continuum opinion dynamics
This paper investigates the asymptotic behavior of some common opinion
dynamic models in a continuum of agents. We show that as long as the
interactions among the agents are symmetric, the distribution of the agents'
opinion converges. We also investigate whether convergence occurs in a stronger
sense than merely in distribution, namely, whether the opinion of almost every
agent converges. We show that while this is not the case in general, it becomes
true under plausible assumptions on inter-agent interactions, namely that
agents with similar opinions exert a non-negligible pull on each other, or that
the interactions are entirely determined by their opinions via a smooth
function.Comment: 28 pages, 2 figures, 3 file
Distributed Evaluation and Convergence of Self-Appraisals in Social Networks
We consider in this paper a networked system of opinion dynamics in
continuous time, where the agents are able to evaluate their self-appraisals in
a distributed way. In the model we formulate, the underlying network topology
is described by a rooted digraph. For each ordered pair of agents , we
assign a function of self-appraisal to agent , which measures the level of
importance of agent to agent . Thus, by communicating only with her
neighbors, each agent is able to calculate the difference between her level of
importance to others and others' level of importance to her. The dynamical
system of self-appraisals is then designed to drive these differences to zero.
We show that for almost all initial conditions, the trajectory generated by
this dynamical system asymptotically converges to an equilibrium point which is
exponentially stable
- …