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

Optimizing Opinions with Stubborn Agents

We consider the problem of optimizing the placement of stubborn agents in a social network in order to maximally influence the population. We assume the network contains stubborn users whose opinions do not change, and non-stubborn users who can be persuaded. We further assume the opinions in the network are in an equilibrium that is common to many opinion dynamics models, including the well-known DeGroot model. We develop a discrete optimization formulation for the problem of maximally shifting the equilibrium opinions in a network by targeting users with stubborn agents. The opinion objective functions we consider are the opinion mean, the opinion variance, and the number of individuals whose opinion exceeds a fixed threshold. We show that the mean opinion is a monotone submodular function, allowing us to find a good solution using a greedy algorithm. We find that on real social networks in Twitter consisting of tens of thousands of individuals, a small number of stubborn agents can non-trivially influence the equilibrium opinions. Furthermore, we show that our greedy algorithm outperforms several common benchmarks. We then propose an opinion dynamics model where users communicate noisy versions of their opinions, communications are random, users grow more stubborn with time, and there is heterogeneity is how users' stubbornness increases. We prove that under fairly general conditions on the stubbornness rates of the individuals, the opinions in this model converge to the same equilibrium as the DeGroot model, despite the randomness and user heterogeneity in the model.Comment: 40 pages, 11 figure

Pattern formation in individual-based systems with time-varying parameters

We study the patterns generated in finite-time sweeps across symmetry-breaking bifurcations in individual-based models. Similar to the well-known Kibble-Zurek scenario of defect formation, large-scale patterns are generated when model parameters are varied slowly, whereas fast sweeps produce a large number of small domains. The symmetry breaking is triggered by intrinsic noise, originating from the discrete dynamics at the micro-level. Based on a linear-noise approximation, we calculate the characteristic length scale of these patterns. We demonstrate the applicability of this approach in a simple model of opinion dynamics, a model in evolutionary game theory with a time-dependent fitness structure, and a model of cell differentiation. Our theoretical estimates are confirmed in simulations. In further numerical work, we observe a similar phenomenon when the symmetry-breaking bifurcation is triggered by population growth.Comment: 16 pages, 9 figures. Published version. Corrected missing appendix link from previous versio

Eminence Grise Coalitions: On the Shaping of Public Opinion

We consider a network of evolving opinions. It includes multiple individuals with first-order opinion dynamics defined in continuous time and evolving based on a general exogenously defined time-varying underlying graph. In such a network, for an arbitrary fixed initial time, a subset of individuals forms an eminence grise coalition, abbreviated as EGC, if the individuals in that subset are capable of leading the entire network to agreeing on any desired opinion, through a cooperative choice of their own initial opinions. In this endeavor, the coalition members are assumed to have access to full profile of the underlying graph of the network as well as the initial opinions of all other individuals. While the complete coalition of individuals always qualifies as an EGC, we establish the existence of a minimum size EGC for an arbitrary time-varying network; also, we develop a non-trivial set of upper and lower bounds on that size. As a result, we show that, even when the underlying graph does not guarantee convergence to a global or multiple consensus, a generally restricted coalition of agents can steer public opinion towards a desired global consensus without affecting any of the predefined graph interactions, provided they can cooperatively adjust their own initial opinions. Geometric insights into the structure of EGC's are given. The results are also extended to the discrete time case where the relation with Decomposition-Separation Theorem is also made explicit.Comment: 35 page