59 research outputs found
How friends and non-determinism affect opinion dynamics
The Hegselmann-Krause system (HK system for short) is one of the most popular
models for the dynamics of opinion formation in multiagent systems. Agents are
modeled as points in opinion space, and at every time step, each agent moves to
the mass center of all the agents within unit distance. The rate of convergence
of HK systems has been the subject of several recent works. In this work, we
investigate two natural variations of the HK system and their effect on the
dynamics. In the first variation, we only allow pairs of agents who are friends
in an underlying social network to communicate with each other. In the second
variation, agents may not move exactly to the mass center but somewhere close
to it. The dynamics of both variants are qualitatively very different from that
of the classical HK system. Nevertheless, we prove that both these systems
converge in polynomial number of non-trivial steps, regardless of the social
network in the first variant and noise patterns in the second variant.Comment: 14 pages, 3 figure
Inertial Hegselmann-Krause Systems
We derive an energy bound for inertial Hegselmann-Krause (HK) systems, which
we define as a variant of the classic HK model in which the agents can change
their weights arbitrarily at each step. We use the bound to prove the
convergence of HK systems with closed-minded agents, which settles a conjecture
of long standing. This paper also introduces anchored HK systems and show their
equivalence to the symmetric heterogeneous model
Differential Game Strategies for Social Networks with Self-Interested Individuals
A social network population engages in collective actions as a direct result
of forming a particular opinion. The strategic interactions among the
individuals acting independently and selfishly naturally portray a
noncooperative game. Nash equilibrium allows for self-enforcing strategic
interactions between selfish and self-interested individuals. This paper
presents a differential game approach to the opinion formation problem in
social networks to investigate the evolution of opinions as a result of a Nash
equilibrium. The opinion of each individual is described by a differential
equation, which is the continuous-time Hegselmann-Krause model for opinion
dynamics with a time delay in input. The objective of each individual is to
seek optimal strategies for her own opinion evolution by minimizing an
individual cost function. Two differential game problems emerge, one for a
population that is not stubborn and another for a population that is stubborn.
The open-loop Nash equilibrium actions and their associated opinion
trajectories are derived for both differential games using Pontryagin's
principle. Additionally, the receding horizon control scheme is used to
practice feedback strategies where the information flow is restricted by fixed
and complete social graphs as well as the second neighborhood concept. The game
strategies were executed on the well-known Zachary's Karate Club social
network. The resulting opinion trajectories associated with the game strategies
showed consensus, polarization, and disagreement in final opinions.Comment: Journal submission under review. arXiv admin note: substantial text
overlap with arXiv:2310.0309
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
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