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

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

An improved energy argument for the Hegselmann-Krause model

We show that the freezing time of the $d$-dimensional Hegselmann-Krause model is $O(n^4)$ where $n$ is the number of agents. This improves the best known upper bound whenever $d\geq 2$

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

Consensus Strikes Back in the Hegselmann-Krause Model of Continuous Opinion Dynamics Under Bounded Confidence

The agent-based bounded confidence model of opinion dynamics of Hegselmann and Krause (2002) is reformulated as an interactive Markov chain. This abstracts from individual agents to a population model which gives a good view on the underlying attractive states of continuous opinion dynamics. We mutually analyse the agent-based model and the interactive Markov chain with a focus on the number of agents and onesided dynamics. Finally, we compute animated bifurcation diagrams that give an overview about the dynamical behavior. They show an interesting phenomenon when we lower the bound of confidence: After the first bifurcation from consensus to polarisation consensus strikes back for a while.Continuous Opinion Dynamics, Bounded Confidence, Interactive Markov Chain, Bifurcation, Number of Agents, Onesided Dynamics