Title: Macroscopic limits of collective dynamics with time-varying weights
We introduce 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 the agent’s weight. The weights evolve in time and their equations are coupled with the opinions’ evolution. We exhibit different kinds of long-term behavior, such as emergence of a single leader and emergence of two co-leaders.
We then focus on two different macroscopic limits of this microscopic system. The so-called “graph limit” translates the concept of indices to an infinite-dimensional system, and can be derived even when the microscopic system does not preserve indistinguishability. When indistinguishability is preserved, we derive the mean-field limit, and obtain a transport equation with source, where the transport term corresponds to the opinion dynamics, and the source term comes from the weight redistribution among the agents. We show the convergence of the microscopic model to both the graph limit equation and the mean-field limit equation, and show the subordination of the mean-field equation to the graph limit one. All results are illustrated with numerical simulations.