arXiv subjects

Jan Lorenz

Publications and source records attributed to Jan Lorenz.

23 records · Page 2Linked to original sources

arXiv2007

Fostering Consensus in Multidimensional Continuous Opinion Dynamics under Bounded Confidence

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.

physics.soc-ph↗

arXiv2007

Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics background

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. Variations of this result have been achieved independently in Lorenz 2005, Moreau 2005 and Hendrickx 2005. We present briefly the opinion dynamics context, discuss the relations to infinite products where infinity is to the right (inhomogeneous Markov processes) and present a small improvement and sketch another.

math.OC↗

arXiv2007

A stabilization theorem for dynamics of continuous opinions

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.

physics.soc-ph↗

arXiv2007

Continuous opinion dynamics of multidimensional allocation problems under bounded confidence: More dimensions lead to better chances for consensus

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 $\eps$. 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.

physics.soc-ph↗

arXiv2006

Convergence in inhomogeneous consensus processes with positive diagonals

We present a results about convergence of products of row-stochastic matrices which are infinite to the left and all have positive diagonals. This is regarded as in inhomogeneous consensus process where confidence weights may change in every time step but where each agent has a little bit of self confidence. The positive diagonal leads to a fixed zero pattern in certain subproducts of the infinite product. We discuss the use of the joint spectral radius on the set of the evolving subproducts and conditions on the subprodutcs to ensure convergence of parts of the infinite product to fixed rank-1-matrices on the diagonal. If the positive minimum of each matrix is uniformly bounded from below the boundedness of the length of intercommunication intervals is important to ensure convergence. We present a small improvement. A slow increase as quick as log(log(t)) in the length of intercommunication intervals is acceptable.

math.OC↗