161 research outputs found
On Convergence Rate of Scalar Hegselmann-Krause Dynamics
In this work, we derive a new upper bound on the termination time of the
Hegselmann-Krause model for opinion dynamics. Using a novel method, we show
that the termination rate of this dynamics happens no longer than
which improves the best known upper bound of by a factor of .Comment: 5 pages, 2 figures, submitted to The American Control Conference,
Sep. 201
On Endogenous Random Consensus and Averaging Dynamics
Motivated by various random variations of Hegselmann-Krause model for opinion
dynamics and gossip algorithm in an endogenously changing environment, we
propose a general framework for the study of endogenously varying random
averaging dynamics, i.e.\ an averaging dynamics whose evolution suffers from
history dependent sources of randomness. We show that under general assumptions
on the averaging dynamics, such dynamics is convergent almost surely. We also
determine the limiting behavior of such dynamics and show such dynamics admit
infinitely many time-varying Lyapunov functions
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
Generalized Opinion Dynamics from Local Optimization Rules
We study generalizations of the Hegselmann-Krause (HK) model for opinion
dynamics, incorporating features and parameters that are natural components of
observed social systems. The first generalization is one where the strength of
influence depends on the distance of the agents' opinions. Under this setup, we
identify conditions under which the opinions converge in finite time, and
provide a qualitative characterization of the equilibrium. We interpret the HK
model opinion update rule as a quadratic cost-minimization rule. This enables a
second generalization: a family of update rules which possess different
equilibrium properties. Subsequently, we investigate models in which a external
force can behave strategically to modulate/influence user updates. We consider
cases where this external force can introduce additional agents and cases where
they can modify the cost structures for other agents. We describe and analyze
some strategies through which such modulation may be possible in an
order-optimal manner. Our simulations demonstrate that generalized dynamics
differ qualitatively and quantitatively from traditional HK dynamics.Comment: 20 pages, under revie
The Hegselmann-Krause dynamics on the circle converge
We consider the Hegselmann-Krause dynamics on a one-dimensional torus and
provide the first proof of convergence of this system. The proof requires only
fairly minor modifications of existing methods for proving convergence in
Euclidean space.Comment: 9 pages, 2 figures. Version 2: A small error in the proof of Theorem
1.1 is corrected and an acknowledgement added. Bibliography update
Opinion Dynamics in Heterogeneous Networks: Convergence Conjectures and Theorems
Recently, significant attention has been dedicated to the models of opinion
dynamics in which opinions are described by real numbers, and agents update
their opinions synchronously by averaging their neighbors' opinions. The
neighbors of each agent can be defined as either (1) those agents whose
opinions are in its "confidence range," or (2) those agents whose "influence
range" contain the agent's opinion. The former definition is employed in
Hegselmann and Krause's bounded confidence model, and the latter is novel here.
As the confidence and influence ranges are distinct for each agent, the
heterogeneous state-dependent interconnection topology leads to a
poorly-understood complex dynamic behavior. In both models, we classify the
agents via their interconnection topology and, accordingly, compute the
equilibria of the system. Then, we define a positive invariant set centered at
each equilibrium opinion vector. We show that if a trajectory enters one such
set, then it converges to a steady state with constant interconnection
topology. This result gives us a novel sufficient condition for both models to
establish convergence, and is consistent with our conjecture that all
trajectories of the bounded confidence and influence models eventually converge
to a steady state under fixed topology.Comment: 22 pages, Submitted to SIAM Journal on Control and Optimization
(SICON
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