Interacting particle systems for opinion dynamics: the Deffuant model and some generalizations

Doctoral thesis, 2016

In the field of sociophysics, concepts and techniques taken from statistical physics are used to model and investigate some social and political behavior of a large group of humans: their social network is given by a simple graph and neighboring individuals meet and interact in pairs or small groups. Although most of the established models feature rather simple microscopic interaction rules, the macroscopic long-time behavior of the collective often eludes an analytical treatment due to the complexity, which stems from the interaction of the large system as a whole. An important class of models in the area of opinion dynamics is the one based on the principle of bounded confidence: Individuals hold and share opinions with others in random encounters. Their mutual influence will lead to updated opinions approaching a compromise, but only if the distance of opinions was not too large in the first place. A popular representative of this class is the model, which was introduced by Deffuant et al. in 2000: Neighboring individuals meet pairwise and symmetrically move towards the average of the two involved opinions if their difference does not exceed a given threshold. In the first paper of this thesis, we study the Deffuant model with real-valued opinions on integer lattices, using geometric and probabilistic tools as well as concepts from statistical physics. These proved to be very effective in the analysis of the model on the integer lattice in dimension 1, i.e. the two-sidedly infinite path Z, and could be adapted to give at least partial results for the lattice in higher dimensions as well as infinite percolation clusters. In papers 2 and 3, we stay on Z but consider a generalization of the model to higher-dimensional opinion spaces, namely vectors and absolutely continuous probability measures, as well as to more general metrics than the Euclidean, used to measure the distance between two opinions. The last appended paper deals with “water transport on graphs”, a new combinatorial optimization problem related to the possible spectrum of opinions for a fixed individual given an initial opinion configuration. We show that on finite graphs, the problem is NP-hard in general and prove a dichotomy that is partly responsible for the fact that our methods used in the analysis of the Deffuant model are less effective on the integer lattice Z^d; d>=2: If the initial values are i.i.d. and bounded, the supremum of values at a fixed vertex – achievable with help of pairwise interactions as in the Deffuant model – depends non-trivially on the initial configuration both for finite graphs and Z, while it a.s. equals the essential supremum of the marginal distribution on higher-dimensional lattices.