Narrow Results

In this paper, we study a model of opinion spreading and consensus formation on a square lattice. Given a two-dimensional square lattice with periodic boundary condition, a total fraction p of the sites are occupied by agents. We take into account the drift ability of the focal agents whose opinions' difference is too large. The dynamics of the model is governed by the following two rules. At each time step, the agents (or individuals) can exchange their opinions with those of their neighbors if the difference between them is smaller than the "bounded confidence" ∊; Otherwise, one can move to a vacancy location if permitted (i.e., there are any empty sites existing among the focal agent's neighborhood). The distribution of opinions over the population evolves toward either a consensus state that all individuals share the same opinion or polarization and fragmentation states where two or more opinions can coexist. The statistical properties of this final state vary considerably as the parameters p and ∊ change. It is shown that, when dispersal abilities of the individuals are considered, the bounded confidence which is required to attain consensus can be remarkably decreased.

We study the interweaving of deterministic and stochastic mechanisms in opinion dynamics. Based on the three-choice Galam model, we introduce a theoretical local-mode featuring with a nearest-neighbor network structure to investigate the effects of local connections. We present both numerical and analytical results for the local and the original global modes. We find that operating in the local mode leads to a much slower but richer process to achieve the opinion consensus. It is shown that in the local-mode dynamics, the stochasticity becomes more effective while the correlations between the initial configurations and final results turn out to be less predictive.

We develop a consensus model for studying opinion dynamics in weighted networks and investigate the effects of both the distribution of weights and the correlations between weight and degree on dynamical behaviors of opinion formation. Our results suggest that a global consensus in the weighted networks to reach is more difficult than that in unweighted network, and strongly depends on the heterogeneity of connection strengths. In addition, in the weighted network with very large heterogeneity of connection strengths, only single macroscopic opinion cluster can be formed, which differs from the behavior in the unweighted network where the next-largest macroscopic opinion cluster may exist.

There are numerous examples of societies with extremely stable mix of contrasting opinions. We argue that this stability is a result of an interplay between society network topology adjustment and opinion changing processes. To support this position we present a computer model of opinion formation based on some novel assumptions, designed to bring the model closer to social reality. In our model, the agents, in addition to changing their opinions due to influence of the rest of society and external propaganda, have the ability to modify their social network, forming links with agents sharing the same opinions and cutting the links with those they disagree with. To improve the model further we divide the agents into "fanatics" and "opportunists," depending on how easy it is to change their opinions. The simulations show significant differences compared to traditional models, where network links are static. In particular, for the dynamical model where inter-agent links are adjustable, the final network structure and opinion distribution is shown to resemble real world observations, such as social structures and persistence of minority groups even when most of the society is against them and the propaganda is strong.

In the consensus formation dynamics, the effect of leaders and interventions have been widely studied for it has many applications such as in politics and commerce. However, the problem is how to know if it is necessary for one to make an intervention. In this paper, we theoretically propose a method for predicting the tendency and final state of collective opinion. By giving each agent a conviction c_i which measures the ability to insist on his opinion, we present an opinion formation model in which agents with high convictions naturally show up properties of the opinion leaders. Results reveal that, although each agent initially gets an opinion evenly distributed in the range [-1, 1], the collective opinion of the steady-state may deviate to the positive or negative direction because of the initial bias of the leaders' opinions. We further get the correlation coefficient of the linear relationship between the collective opinion and the initial bias according to both the experimental and theoretical analysis. Thus, we could predict the final state at the very beginning of the dynamic only if we get the opinions of a small portion of the population. The prediction would afford us more time and opportunities to make reactions and interventions.

We introduce a novel first-order stochastic swarm intelligence (SI) model in the spirit of consensus formation models, namely a consensus-based optimization (CBO) algorithm, which may be used for the global optimization of a function in multiple dimensions. The CBO algorithm allows for passage to the mean-field limit, which results in a nonstandard, nonlocal, degenerate parabolic partial differential equation (PDE). Exploiting tools from PDE analysis we provide convergence results that help to understand the asymptotic behavior of the SI model. We further present numerical investigations underlining the feasibility of our approach.

In this paper, we provide an analytical framework for investigating the efficiency of a consensus-based model for tackling global optimization problems. This work justifies the optimization algorithm in the mean-field sense showing the convergence to the global minimizer for a large class of functions. Theoretical results on consensus estimates are then illustrated by numerical simulations where variants of the method including nonlinear diffusion are introduced.

In this paper, we investigate the asymptotic formation of consensus for several classes of time-dependent cooperative graphon dynamics. After motivating the use of this type of macroscopic models to describe multi-agent systems, we adapt the classical notion of scrambling coefficient to this setting, and leverage it to establish sufficient conditions ensuring the exponential convergence to consensus with respect to the $L^{\infty }$-norm topology. We then shift our attention to consensus formation expressed in terms of the $L^2$-norm, and prove three different results for symmetric, balanced and strongly connected topologies, which involve a suitable generalisation of the notion of algebraic connectivity to this infinite-dimensional framework. We then show that, just as in the finite-dimensional setting, the notion of algebraic connectivity that we propose encodes information about the connectivity properties of the underlying interaction topology. We finally use the corresponding results to shed some light on the relation between $L^2$- and $L^{\infty }$-consensus formation, and illustrate our contributions by a series of numerical simulations.

We study the role of hierarchical structures in a simple model of collective consensus formation based on the bounded confidence model with continuous individual opinions. For the particular variation of this model considered in this paper, we assume that a bias toward an extreme opinion is introduced whenever two individuals interact and form a common decision. As a simple proxy for hierarchical social structures, we introduce a two-step decision making process in which in the second step groups of like-minded individuals are replaced by representatives once they have reached local consensus, and the representatives in turn form a collective decision in a downstream process. We find that the introduction of such a hierarchical decision making structure can improve consensus formation, in the sense that the eventual collective opinion is closer to the true average of individual opinions than without it. In particular, we numerically study how the size of groups of like-minded individuals being represented by delegate individuals affects the impact of the bias on the final population-wide consensus. These results are of interest for the design of organizational policies and the optimization of hierarchical structures in the context of group decision making.