This paper introduces an augmented model for first-order opinion dynamics, in which a weight of influence is attributed to each agent. Each agent’s influence on another agent’s opinion is then proportional not only to the classical interaction function, but also to its weight. The weights evolve in time and their equations are coupled with the opinions’ evolution. We show that the well-known conditions for convergence to consensus can be generalized to this framework. In the case of interaction functions with bounded support, we show that constant weights lead to clustering with conditions similar to those of the classical model. Four specific models are designed by prescribing a specific weight dynamics, then the convergence of the opinions and the evolution of the weights for each of them are studied. We prove the existence of different long-term behaviors, such as emergence of a single leader and emergence of two co-leaders. Then we illustrate them via numerical simulations. Last, a statistical analysis is provided for the speed of convergence to consensus and for the clustering behavior of each model, together with a comparison to the classical opinion dynamics with constant equal weights.
Mean-field and graph limits for collective dynamics models with time-varying weights
