Equivalent bounded confidence processes
Sascha Kurz
TL;DR
The paper tackles the infinite variety of Hegselmann–Krause bounded confidence processes on the real line by introducing finite equivalence classes of dynamics induced by threshold choices. It develops a comprehensive framework using influence graphs and linear programming to realize and compare possible graph sequences, enabling exact or bounded analyses of freezing time and fragmentation for small agent counts. Key contributions include multiple notions of equivalence (affine, ε-, and graph-equivalence), a transition-graph LP formulation to decide realizability of dynamics, and precise results for maximal freezing time F(n) and maximal fragmentation S(n) with exact values for small n and asymptotic behavior. This approach provides a principled, computationally tractable method to exhaustively explore BC-processes and sheds light on the combinatorial structures (e.g., interval and unit interval graphs) that govern long-term behavior.
Abstract
In the bounded confidence model the opinions of a set of agents evolve over discrete time steps. In each round an agent averages the opinion of all agents whose opinions are at most a certain threshold apart. Here we assume that the opinions of the agents are elements of the real line. The details of the dynamics are determined by the initial opinions of the agents, i.e. a starting configuration, and the mentioned threshold -- both allowing uncountable infinite possibilities. Recently it was observed that for each starting configuration the set of thresholds can be partitioned into a finite number of intervals such that the evolution of opinions does not depend on the precise value of the threshold within one of the intervals. So, we may say that, given a starting configuration of initial opinions, there is only a finite number of equivalence classes of bounded confidence processes (and an algorithm to compute them). Here we systematically study different notions of equivalence. In our widest notion we can also get rid of the initial starting configuration and end up with a finite number of equivalent bounded confidence processes for each given (finite) number of agents. This allows to precisely study the occurring phenomena for small numbers of agents without the jeopardy of missing interesting cases by performing numerical experiments. We exemplarily study the freezing time, i.e. number of time steps needed until the process stabilizes, and the degree of fragmentation, i.e. the number of different opinions that survive once the process has reached its final state.