Consensus Strikes Back in the Hegselmann-Krause Model of Continuous Opinion Dynamics Under Bounded Confidence

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.Continuous Opinion Dynamics, Bounded Confidence, Interactive Markov Chain, Bifurcation, Number of Agents, Onesided Dynamics

Order preservation in a generalized version of Krause's opinion dynamics model

Krause's model of opinion dynamics has recently been the object of several studies, partly because it is one of the simplest multi-agent systems involving position-dependent changing topologies. In this model, agents have an opinion represented by a real number and they update it by averaging those agent opinions distant from their opinion by less than a certain interaction radius. Some results obtained on this model rely on the fact that the opinion orders remain unchanged under iteration, a property that is consistent with the intuition in models with simultaneous updating on a fully connected communication topology. Several variations of this model have been proposed. We show that some natural variations are not order preserving and therefore cause potential problems with the theoretical analysis and the consistence with the intuition. We consider a generic version of Krause's model parameterized by an influence function that encapsulates most of the variations proposed in the literature. We then derive a necessary and sufficient condition on this function for the opinion order to be preserved.Comment: 10 pages, 6 figures, 13 eps file

About the Power to Enforce and Prevent Consensus by Manipulating Communication Rules

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 \eps. A high \eps leads to consensus, while a lower \eps leads to a fragmentation into several opinion clusters. We drop the random encounter assumption and ask: How small may \eps be such that consensus is still possible with a certain communication plan for the entire group? Mathematical analysis shows that \eps may be significantly smaller than in the random pairwise case. On the other hand we ask: How large may \eps be such that preventing consensus is still possible? In answering this question we prove Fortunato's simulation result that consensus cannot be prevented for \eps>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.Comment: 21 pages, 6 figure

How groups can foster consensus: The case of local cultures

A local culture denotes a commonly shared behaviour within a cluster of firms. Similar to social norms or conventions, it is an emergent feature resulting from the firms' interaction in an economic network. To model these dynamics, we consider a distributed agent population, representing e.g. firms or individuals. Further, we build on a continuous opinion dynamics model with bounded confidence ($\epsilon$), which assumes that two agents only interact if differences in their behaviour are less than $\epsilon$. Interaction results in more similarity of behaviour, i.e. convergence towards a common mean. This framework is extended by two major concepts: (i) The agent's in-group consisting of acquainted interaction partners is explicitly taken into account. This leads to an effective agent behaviour reflecting that agents try to continue to interact with past partners and thus to keep sufficiently close to them. (ii) The in-group network structure changes over time, as agents can form new links to other agents with sufficiently close effective behaviour or delete links to agents no longer close in behaviour. Thus, our model provides a feedback mechanism between the agents' behaviour and their in-group structure. Studying its consequences by means of agent-based computer simulations, we find that for narrow-minded agents (low $\epsilon$) the additional feedback helps to find consensus more often, whereas for open-minded agents (high $\epsilon$) this does not hold. This counterintuitive result is explained by simulations of the network evolution