Abstract: Models of continuous opinion dynamics under bounded confidence show a sharp transition between a consensus and a polarization phase at a critical global bound of confidence. In this paper, heterogeneous bounds of confidence are studied. The surprising result is that a society of agents with two different bounds of confidence (open-minded and closed-minded agents) can find consensus even when both bounds of confidence are significantly below the critical bound of confidence of a homogeneous society.

The phenomenon is shown by examples of agent-based simulation and by numerical computation of the time evolution of the agents density. The result holds for the bounded confidence model of Deffuant, Weisbuch and others (Weisbuch, G. et al; Meet, discuss, and segregate!, Complexity, 2002, 7, 55--63), as well as for the model of Hegselmann and Krause (Hegselmann, R., Krause, U.; Opinion Dynamics and Bounded Confidence, Models, Analysis and Simulation, Journal of Artificial Societies and Social Simulation, 2002, 5, 2).

Thus, given an average level of confidence, diversity of bounds of confidence enhances the chances for consensus. The drawback of this enhancement is that opinion dynamics becomes suspect to severe drifts of clusters, where open-minded agents can pull closed-minded agents towards another cluster of closed-minded agents. A final consensus might thus not lie in the center of the opinion interval as it happens for uniform initial opinion distributions under homogeneous bounds of confidence. It can be located at extremal locations. This is demonstrated by example.

This also show that the extension to heterogeneous bounds of confidence enriches the complexity of the dynamics tremendously.

Abstract: In this paper histograms of user ratings for movies (1,...,10) are analysed. The evolving stabilised shapes of histograms follow the rule that all are either double- or triple-peaked. Moreover, at most one peak can be on the central bins 2,...,9 and the distribution in these bins looks smooth `Gaussian-like' while changes at the extremes (1 and 10) often look abrupt. It is shown that this is well approximated under the assumption that histograms are confined and discretised probability density functions of Lévy skew alpha-stable distributions. These distributions are the only stable distributions which could emerge due to a generalized central limit theorem from averaging of various independent random avriables as which one can see the initial opinions of users. Averaging is also an appropriate assumption about the social process which underlies the process of continuous opinion formation. Surprisingly, not the normal distribution achieves the best fit over histograms obseved on the web, but distributions with fat tails which decay as power-laws with exponent -(1+alpha) (alpha=4/3). The scale and skewness parameters of the Levy skew alpha-stable distributions seem to depend on the deviation from an average movie (with mean about 7.6). The histogram of such an average movie has no skewness and is the most narrow one. If a movie deviates from average the distribution gets broader and skew. The skewness pronounces the deviation. This is used to construct a one parameter fit which gives some evidence of universality in processes of continuous opinion dynamics about taste.

Abstract: We introduce a general framework for models of cascade and contagion processes on networks, to identify their commonalities and differences. In particular, models of social and financial cascades, as well as the fiber bundle model, the voter model, and models of epidemic spreading are recovered as special cases. To unify their description, we define the net fragility of a node, which is the difference between its fragility and the threshold that determines its failure. Nodes fail if their net fragility grows above zero and their failure increases the fragility of neighbouring nodes, thus possibly triggering a cascade. In this framework, we identify three classes depending on the way the fragility of a node is increased by the failure of a neighbour. At the microscopic level, we illustrate with specific examples how the failure spreading pattern varies with the node triggering the cascade, depending on its position in the network and its degree. At the macroscopic level, systemic risk is measured as the final fraction of failed nodes, $X^, and for each of the three classes we derive a recursive equation to compute its value. The phase diagram of $X^ as a function of the initial conditions, thus allows for a prediction of the systemic risk as well as a comparison of the three different model classes. We could identify which model class lead to a first-order phase transition in systemic risk, i.e. situations where small changes in the initial conditions may lead to a global failure. Eventually, we generalize our framework to encompass stochastic contagion models. This indicates the potential for further generalizations.

Abstract: Assume two different communities each of which maintain their respective opinions mainly because of the weak interaction between the two communities. In such a case, it is an interesting problem to find the necessary strength of inter-community interaction in order for the two communities to reach a consensus. In this paper, the Information Accumulation System (IAS) model is applied to investigate the problem. With the application of the IAS model, the opinion dynamics of the two-community problem is found to belong to a wider class of two-species problems appearing in population dynamics or in competition of two languages, for all of which the governing equations can be described in terms of coupled logistic maps. A concept of tipping diffusivity is defined as a maximum inter-community interaction that the two communities can stay in different opinions. For a problem with simple community structure and homogeneous individuals, the tipping diffusivity is calculated theoretically. As a conclusion of the paper, the convergence of the two communities to the same value is less possible the more overall interaction, intra-community and inter-community, takes place. This implies, for example, that the increase in the interaction between individuals caused by the development of the modern communication tools, such as facebook and twitter, does not necessarily improve the tendency toward global convergence between different communities. If the number of internal links increases, the number of inter-community links need to be increased even more, in order for consensus to be the only stable attractor.

Abstract: Social consensus is important for society. Sometimes the success of society depends on a consensus (e.g. the decision to pay taxes or to commit to the constitution). Examples for continuous opinion dynamics are discussions about tax rates or budget plan proposals for governments investments. Another example is a commission of experts which should reach a estimate about a certain issue, e.g. the tax revenues of the next year. In all these situations we got a group of agents which should reach a common agreement either for reaching a good approximation to the truth but on the other hand for the reason, that reaching consensus is a good in itself.

From social judgment theory and experiments we know that humans either tend to agreement with others for normative and informational reasons but on the other hand have bounded confidence against others with differing opinions.

In a framework of models of continuous opinion dynamics we ask, which structural conditions foster the achievement of consensus? We present evidence by simulation that bringing more issues in does, but only if the issues are under budget constraints. Further, the installation of meetings where everyone hears all opinions has a better impact than relying on gossip.

Abstract: We study the mean field approximation of a recent model of cascades on networks relevant to the investigation of systemic risk control in financial networks. In the model, the hypothesis of a trend reinforcement in the stochastic process describing the fragility of the nodes, induces a trade-off in the systemic risk with respect to the density of the network. Increasing the average link density, the network is first less exposed to systemic risk, while above an intermediate value the systemic risk increases. This result offers a simple explanation for the emergence of instabilities in financial systems that get increasingly interwoven. In this paper, we study the dynamics of the probability density function of the average fragility. This converges to a unique stable distribution which can be computed numerically and can be used to estimate the systemic risk as a function of the parameters of the model.

Abstract: A new theorem on conditions for convergence to consensus of a multiagent time-dependent time-discrete dynamical system is presented. The theorem is build up on the notion of averaging maps. We compare this theorem to Moreau's Theorem and his proposed set-valued Lyapunov theory (IEEE Transactions on Automatic Control, vol. 50, no. 2, 2005). We give examples that point out differences of approaches including examples where Moreau's theorem is not applicable but ours is. Further on, we give examples that demonstrate that the theory of convergence to consensus is still not complete.

Abstract: The opinion dynamics model introduced by Deffuant and Weisbuch as well as the one by Hegselmann and Krause are rather similar. In both models individuals are assumed to have opinions about an issue, they meet and discuss, and they may adapt their opinions towards the other agents` opinions or may ignore each other if their positions are too different. Both models differ with respect to the number of peers they meet at once. Furthermore the model by Deffuant and Weisbuch has a convergence parameter that controls how fast agents adapt their opinions. By defining the reversed parameter as self-support we can extend the applicability of this parameter to scenarios with more than one interaction partner. We investigate the effects of changing the number of peers met at once, which is done for different population sizes, and the effects of changing the self-support. For describing the dynamics we look at different statistics, i.e. number of cluster, number of major clusters, and Gini coefficient.

Abstract: Models of continuous opinion dynamics under bounded confidence have been presented independently by Krause and Hegselmann and by Deffuant et al in 2000. They have raised a fair amount of attention in the communities of social simulation, sociophysics and complexity science. The researchers working on it come from disciplines as physics, mathematics, computer science, social psychology and philosophy.

Agents hold continuous opinions which they can gradually adjust if they hear the opinions of others. The idea of bounded confidence is that agents only interact if they are close in opinion to each other. Usually, the models are analyzed with agent-based simulations in a Monte-Carlo style, but they can also be reformulated on the agent's density in the opinion space in a master-equation style. This paper is to present the agent-based and density-based modeling frameworks including the cases of multidimensional opinions and heterogeneous bounds of confidence; second, to give the bifurcation diagrams of cluster configuration in the homogeneous model with uniformly distributed initial opinions; third to review the several extensions and the evolving phenomena which have been studied so far; and fourth to state some basic open questions.

Abstract: We present two models of continuous opinion dynamics under bounded confidence which are representable as nonnegative discrete dynamical systems, namely the Hegselmann-Krause model (Hegselmann and Krause, Journal of Artificial Societies and Social Simulation 5(3), 2002) and the Deffuant-Weisbuch model (Deffuant et al, Advances in Complex Systems, 3, 2000). We fully characterize the set of fixed points for both models. They are identical. Further on, we present reformulations of both models on the more general level of densities of agents in the opinion space as interactive Markov chains. We also characterize the sets of fixed points as identical in both models.

Abstract: This thesis is about dynamical systems of agents which perform repeated averaging under bounded confidence. It contributes to questions of their modeling, mathematical analysis and simulation.

Modeling. The main modeling issue is continuous opinion dynamics. This includes dynamics of agents in a political opinion space as well as dynamics of collective motion in swarms of mobile autonomous robots. The existing bounded confidence models of Hegselmann-Krause and Deffuant-Weisbuch are presented under a common general framework. For both models density-based versions are introduced which give additional options for analysis. Surprising phenomena are presented, e.g. fostering consensus by lowering confidence or by introducing heterogeneity.

Mathematics. Conditions for convergence to consensus are derived for systems where dynamics is driven by very generally defined averaging maps. If averaging maps are linear then the fundamental part of the system is a product of row-stochastic matrices infinite two the left. Several conditions, examples and counter-examples for convergence of such products are given. Thereby, we extend a result about convergence to consensus: Intercommunication intervals need not be bounded but may grow very very slow. Finally, the sets of fixed points are characterized for the bounded confidence models.

Simulation. Interesting candidates for this class of systems are the models of opinion dynamics under bounded confidence, which are then analyzed via computer simulation. The density-based modeling approach is used to get a systematic overview for the case when agents' initial opinions are uniformly distributed in the opinion space. Dynamics always converges to situations with clustered opinions. Bifurcation diagrams for attractive clustered states are computed for homogeneous bounds of confidence as well as extended phase diagrams for the consensus transitions in populations with two different levels of confidence. The thesis concludes with some advices on how to foster consensus in continuous opinion dynamics under bounded confidence and a list of open problems.

Abstract: We explore the possibilities of enforcing and preventing consensus in continuous opinion dynamics that result from modifications in the communication rules. We refer to the model of Weisbuch and Deffuant, where n agents adjust their continuous opinions as a result of random pairwise encounters whenever their opinions differ not more than a given bound of confidence epsilon. A high epsilon leads to consensus, while a lower epsilon leads to a fragmentation into several opinion clusters. We drop the random encounter assumption and ask: How small may epsilon be such that consensus is still possible with a certain communication plan for the entire group? Mathematical analysis shows that epsilon may be significantly smaller than in the random pairwise case. On the other hand we ask: How large may epsilon be such that preventing consensus is still possible? In answering this question we prove Fortunato's simulation result that consensus cannot be prevented for epsilon>0.5 for large groups. Next we consider opinion dynamics under different individual strategies and examine their power to increase the chances of consensus. One result is that balancing agents increase chances of consensus, especially if the agents are cautious in adapting their opinions. However, curious agents increase chances of consensus only if those agents are not cautious in adapting their opinions.

Abstract: 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.

Abstract: We study multidimensional continuous opinion dynamics, where opinions are nonnegative vectors which components sum up to one. Examples of such opinions are budgets or other allocation vectors which display a distribution of a fixed amount of ressource to n projects.

We use the opinion dynamics models of Deffuant-Weisbuch and Hegselmann-Krause, which both extend naturally to more dimensional opinions. They both rely on bounded confidence of the agents and differ in their communication regime. We show detailed simulation results regarding n=2,...,8 and the bound of confidence epsilon. Number, location and size of opinion clusters in the stabilized opinion profiles are of interest.

Known differences of both models repeat under higher opinion dimensions: Higher number of clusters and more minor clusters in the Deffuant-Weisbuch model, meta-stable states in the Hegselmann-Krause model. But surprisingly, higher dimensions lead to better chances for a vast majority consensus even for lower bounds of confidence. On the other hand, the number of minority clusters rises with n, too.

Abstract: We present a convergence result for infinite products of stochastic matrices with positive diagonals. We regard infinity of the product to the left. Such a product converges partly to a fixed matrix if the minimal positive entry of each matrix does not converge too fast to zero and if either zero-entries are symmetric in each matrix or the length of subproducts which reach the maximal achievable connectivity is bounded.

Abstract: A stabilization theorem for processes of opinion dynamics is presented. The theorem is applicable to a wide class of models of continuous opinion dynamics based on averaging (like the models of Hegselmann-Krause and Weisbuch-Deffuant). The analysis detects self-confidence as a driving force of stabilization.

Abstract: We reformulate the agent-based opinion dynamics models of Weisbuch-Deffuant and Hegselmann-Krause as interactive Markov chains. So we switch the scope from a finite number of n agents to a finite number of n opinion classes. Thus, we will look at an infinite population distributed to opinion classes instead of agents with real number opinions. The interactive Markov chains show similar dynamical behavior as the agent-based models: stabilization and clustering. Our framework leads to a discrete bifurcation diagram for each model which gives a good view on the driving forces and the attractive states of the system. The analysis shows that the emergence of minor clusters in the Weisbuch-Deffuant model and of meta-stable states with very slow convergence to consensus in the Hegselmann Krause model are intrinsic to the dynamical behavior.

Abstract: In this paper, participatory budgeting (especially in Porto Alegre) is seen as a formal mathematical model of opinion dynamics and social-choice. We relate it to the classical social-choice problem mixed with the ideas of the wisdom of crowds and social consensus. Mathematical Simulation of opinion dynamics are done to analyse the impact of the mechanism on the finding of a social consensus. With these mathematical results, we aim to contribute to the debate about introducing participatory budgeting in Germany and how to extend it in Brazil. The simulation of opinion dynamics leads to the result that giving the crowd more categories to distribute money to has an impact that fosters at least a vast majority of the group to find a social consensus. But this effect slows down so that about eight categories seem to be enough. We conclude by putting our results into the general picture covering the political barriers of implementing participatory budgeting: representation, power balance and institutionalization. And we add some remarks about participatory budgeting as a tool to evolve from technobureaucracy to technodemocracy and about the civil society as a forth force in democracy.

Abstract: A group of m agents is to find a common agreement about a certain issue. Consider this issue to be an n-dimensional vector of real numbers, for example the allocation of a fixed sum of money to n projects. Each agent has an opinion about the allocation, which he may revise. We model the process of opinion formation as a time-discrete dynamical system, in which every agent averages all opinions which are closed to his own opinion. Thus condence structures may change. Mathematical analysis shows that every starting opinion distribution converges to a stable distribution. The driving force of this convergence is self-confidence. Further, we present some simulation results concerning m = 150 and n = 3, which give an insight into the dynamics of multidimensional opinion dynamics under bounded confidence.

Abstract: A group of m agents is to find a common agreement about a certain issue. An opinion is represented by a real number. We assume that each agent changes his opinion by repeated averaging over the opinions of his agents of confidence. An agent i should be an agent of confidence for agent j, if the opinion of agent i lies within the confidence interval of agent j, which is an interval with the opinion of j in the center. This iterated process is called opinion formation under bounded confidence. In this paper we model heterogenity of agents by admitting differences in the sizes of the confidence intervals for each agent. We examine what happens if some of the agents have higher/lower bounds of confidence than the other agents. We focus on how the possibility of reaching a consensus among all agents changes through heterogenity. Interestingly there are both, positive and negative effects. Simulations show that the positive effects might be bigger. In addition, we present some mathematical results about the convergence of this kind of opinion formation.