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Register a new userProbability of consensus in the multivariate Deffuant model on finite connected graphs
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The Deffuant model is a spatial stochastic model for the dynamics of opinions in which individuals are located on a connected graph representing a social network and characterized by a number in the unit interval representing their opinion. The system evolves according to the following averaging procedure: pairs of neighbors interact independently at rate one if and only if the distance between their opinions does not exceed a certain confidence threshold, with each interaction resulting in the neighbors opinions getting closer to each other. All the mathematical results collected so far about this model assume that the individuals are located on the integers. In contrast, we study the more realistic case where the social network can be any finite connected graph. In addition, we extend the opinion space to any bounded convex subset of a normed vector space where the norm is used to measure the level of disagreement or distance between the opinions. Our main result gives a lower bound for the probability of consensus. Interestingly, our proof leads to a universal lower bound that depends on the confidence threshold, the opinion space~(convex subset and norm) and the initial distribution, but not on the size or the topology of the social network.
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During the last decades, quite a number of interacting particle systems have been introduced and studied in the border area of mathematics and statistical physics. Some of these can be seen as simplistic models for opinion formation processes in groups of interacting people. In the one introduced by Deffuant et al. agents, that are neighbors on a given network graph, randomly meet in pairs and approach a compromise if their current opinions do not differ by more than a given threshold value $theta$. We consider the two-sidedly infinite path $mathbb{Z}$ as underlying graph and extend former investigations to a setting in which opinions are given by probability distributions. Similar to what has been shown for finite-dimensional opinions, we observe a dichotomy in the long-term behavior of the model, but only if the initial narrow-mindedness of the agents is restricted.
When it comes to the mathematical modelling of social interaction patterns, a number of different models have emerged and been studied over the last decade, in which individuals randomly interact on the basis of an underlying graph structure and share their opinions. A prominent example of the so-called bounded confidence models is the one introduced by Deffuant et al.: Two neighboring individuals will only interact if their opinions do not differ by more than a given threshold $theta$. We consider this model on the line graph $mathbb{Z}$ and extend the results that have been achieved for the model with real-valued opinions by considering vector-valued opinions and general metrics measuring the distance between two opinion values. Just as in the univariate case, there exists a critical value for $theta$ at which a phase transition in the long-term behavior takes place.
We employ stabilization methods and second order Poincare inequalities to establish rates of multivariate normal convergence for a large class of vectors $(H_s^{(1)},...,H_s^{(m)})$, $s geq 1$, of statistics of marked Poisson processes on $mathbb{R}^d$, $d geq 2$, as the intensity parameter $s$ tends to infinity. Our results are applicable whenever the constituent functionals $H_s^{(i)}$, $iin{1,...,m}$, are expressible as sums of exponentially stabilizing score functions satisfying a moment condition. The rates are for the $d_2$-, $d_3$-, and $d_{convex}$-distances. When we compare with a centered Gaussian random vector, whose covariance matrix is given by the asymptotic covariances, the rates are in general unimprovable and are governed by the rate of convergence of $s^{-1} {rm Cov}( H_s^{(i)}, H_s^{(j)})$, $i,jin{1,...,m}$, to the limiting covariance, shown to be of order $s^{-1/d}$. We use the general results to deduce rates of multivariate normal convergence for statistics arising in random graphs and topological data analysis as well as for multivariate statistics used to test equality of distributions. Some of our results hold for stabilizing functionals of Poisson input on suitable metric spaces.
A finite graph $G$ is said to be {em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $gin G$ of the graph, where $G$ is a group of automorphisms of the graph. In 1985, Cameron and Macpherson determined all finite $(G, 3)$-homogeneous graphs. In this paper, we develop a method for characterising $(G,3)$-connected homogeneous graphs. It is shown that for a finite $(G,3)$-connected homogeneous graph $G=(V, E)$, either $G_v^{G(v)}$ is $2$--transitive or $G_v^{G(v)}$ is of rank $3$ and $G$ has girth $3$, and that the class of finite $(G,3)$-connected homogeneous graphs is closed under taking normal quotients. This leads us to study graphs where $G$ is quasiprimitive on $V$. We determine the possible quasiprimitive types for $G$ in this case and give new constructions of examples for some possible types.
We investigate a model for opinion dynamics, where individuals (modeled by vertices of a graph) hold certain abstract opinions. As time progresses, neighboring individuals interact with each other, and this interaction results in a realignment of opinions closer towards each other. This mechanism triggers formation of consensus among the individuals. Our main focus is on strong consensus (i.e. global agreement of all individuals) versus weak consensus (i.e. local agreement among neighbors). By extending a known model to a more general opinion space, which lacks a central opinion acting as a contraction point, we provide an example of an opinion formation process on the one-dimensional lattice with weak consensus but no strong consensus.
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