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.
In this paper we interpret the Hegselmann-Krause dynamics in terms of utility optimization. There is a significant body of work on utility-maximization in multi-agent settings. @cite_7 considers a large population linear Quadratic Gaussian (LQG) problem where agents interact via coupled cost. There are other works involving control theoretic, game theoretic and optimization based approaches. @cite_1 presents a survey of recent developments. Here we study an engineering problem where the society follows Hegselmann-Krause dynamics and a strategic entity wants to form opinions in its favor. This problem of opinion modulation modification is analogous to marketing and political campaigns.
@cite_1 @cite_7
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We consider linear quadratic Gaussian (LQG) games in large population systems where the agents evolve according to nonuniform dynamics and are coupled via their individual costs. A state aggregation technique is developed to obtain a set of decentralized control laws for the individuals which possesses an epsiv-Nash equilibrium property. A stability property of the mass behavior is established, and the effect of inaccurate population statistics on an isolated agent is also analyzed by variational techniques. We study convergence of the following discrete-time non-linear dynamical system: @math agents are located in Rd and at every time step, each moves synchronously to the average location of all agents within a unit distance of it. This popularly studied system was introduced by Krause to model the dynamics of opinion formation and is often referred to as the Hegselmann-Krause model. We prove the first polynomial time bound for the convergence of this system in arbitrary dimensions. This improves on the bound of nO(n) resulting from a more general theorem of Chazelle [4]. Also, we show a quadratic lower bound and improve the upper bound for one-dimensional systems to O(n3).