8 research outputs found
An improved energy argument for the Hegselmann-Krause model
We show that the freezing time of the -dimensional Hegselmann-Krause model
is where is the number of agents. This improves the best known
upper bound whenever
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
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
A quadratic lower bound for the convergence rate in the one-dimensional Hegselmann-Krause bounded confidence dynamics
Let f_{k}(n) be the maximum number of time steps taken to reach equilibrium by a system of n agents obeying the -dimensional Hegselmann-Krause bounded confidence dynamics. Previously, it was known that \Omega(n) = f_{1}(n) = O(n^3). Here we show that f_{1}(n) = \Omega(n^2), which matches the best-known lower bound in all dimensions k >= 2