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

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 $O(n^3)$ which improves the best known upper bound of $O(n^4)$ by a factor of $n$ .Comment: 5 pages, 2 figures, submitted to The American Control Conference, Sep. 201

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

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

Mean-field sparse Jurdjevic-Quinn control

International audienceWe consider nonlinear transport equations with non-local velocity, describing the time-evolution of a measure, which in practice may represent the density of a crowd. Such equations often appear by taking the mean-field limit of finite-dimensional systems modelling collective dynamics. We first give a sense to dissipativity of these mean-field equations in terms of Lie derivatives of a Lyapunov function depending on the measure. Then, we address the problem of controlling such equations by means of a time-varying bounded control action localized on a time-varying control subset with bounded Lebesgue measure (sparsity space constraint). Finite-dimensional versions are given by control-affine systems, which can be stabilized by the well known Jurdjevic–Quinn procedure. In this paper, assuming that the uncontrolled dynamics are dissipative, we develop an approach in the spirit of the classical Jurdjevic–Quinn theorem, showing how to steer the system to an invariant sublevel of the Lyapunov function. The control function and the control domain are designed in terms of the Lie derivatives of the Lyapunov function, and enjoy sparsity properties in the sense that the control support is small. Finally, we show that our result applies to a large class of kinetic equations modelling multi-agent dynamics

Bounded Confidence under Preferential Flip: A Coupled Dynamics of Structural Balance and Opinions

In this work we study the coupled dynamics of social balance and opinion formation. We propose a model where agents form opinions under bounded confidence, but only considering the opinions of their friends. The signs of social ties -friendships and enmities- evolve seeking for social balance, taking into account how similar agents' opinions are. We consider both the case where opinions have one and two dimensions. We find that our dynamics produces the segregation of agents into two cliques, with the opinions of agents in one clique differing from those in the other. Depending on the level of bounded confidence, the dynamics can produce either consensus of opinions within each clique or the coexistence of several opinion clusters in a clique. For the uni-dimensional case, the opinions in one clique are all below the opinions in the other clique, hence defining a "left clique" and a "right clique". In the two-dimensional case, our numerical results suggest that the two cliques are separated by a hyperplane in the opinion space. We also show that the phenomenon of unidimensional opinions identified by DeMarzo, Vayanos and Zwiebel (Q J Econ 2003) extends partially to our dynamics. Finally, in the context of politics, we comment about the possible relation of our results to the fragmentation of an ideology and the emergence of new political parties.Comment: 8 figures, PLoS ONE 11(10): e0164323, 201

Optimal Opinion Control: The Campaign Problem

Opinion dynamics is nowadays a very common field of research. In this article we formulate and then study a novel, namely strategic perspective on such dynamics: There are the usual normal agents that update their opinions, for instance according the well-known bounded confidence mechanism. But, additionally, there is at least one strategic agent. That agent uses opinions as freely selectable strategies to get control on the dynamics: The strategic agent of our benchmark problem tries, during a campaign of a certain length, to influence the ongoing dynamics among normal agents with strategically placed opinions (one per period) in such a way, that, by the end of the campaign, as much as possible normals end up with opinions in a certain interval of the opinion space. Structurally, such a problem is an optimal control problem. That type of problem is ubiquitous. Resorting to advanced and partly non-standard methods for computing optimal controls, we solve some instances of the campaign problem. But even for a very small number of normal agents, just one strategic agent, and a ten-period campaign length, the problem turns out to be extremely difficult. Explicitly we discuss moral and political concerns that immediately arise, if someone starts to analyze the possibilities of an optimal opinion control.Comment: 47 pages, 12 figures, and 11 table

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

Sociophysics Simulations II: Opinion Dynamics

Individuals have opinions but can change them under the influence of others. The recent models of Sznajd (missionaries), of Deffuant et al. (negotiators), and of Krause and Hegselmann (opportunists) are reviewed here, while the voter and Ising models, Galam's majority rule and the Axelrod multicultural model were dealt with by other lecturers at this 8th Granada Seminar.Comment: 18 pages including 9 figs., for 8th Granada seminar (AIP Conf.Proc.