Collective Behavior in Network-Based Dynamical Systems

Abstract

This thesis investigates the emergence of collective behavior in network-based dynamical systems. By focusing successively on inertial Hegselmann-Krause (HK) systems, noisy HK systems, and network-based iterated learning mechanisms, we are able to uncover deep relations in the system dynamics and resolve open problems of long standing.

We introduce inertial HK systems as variants of the classic HK model in which the agents can change their weights arbitrarily at each step. We derive an energy bound for these systems via an algorithmic proof, which can be interpreted as a time-dependent message-passing protocol designed to track the moving “potentials” of the agents. Building on this relation, we resolve a long-standing open problem: the convergence of HK systems in the presence of closed-minded agents. To investigate the effects of noise on network-based dynamics, we introduce noisy HK systems, which model in a time-continuous framework the tension between two competing forces: the attraction between agents with similar opinions and the diffusion caused by the noise. Using perturbation analysis of the system’s mean-field limiting Fokker-Planck equation, we provide a theoretical explanation for the celebrated 2R conjecture, which states that, for a random initial distribution in a fixed interval, the final configuration consists of clusters separated by a distance of roughly 2R.

In the last part of the thesis, we formalize the iterated learning problem within a network-based dynamical system framework. In its sequential setting, the belief of the learner is found to converge exponentially fast to their prior distribution if the amount of information transferred in each step remains the same. We demonstrate that, by increasing the data transferred at each step logarithmically, the original information can be sustained with arbitrary accuracy. In a social learning setting in which the learners are truth-seeking, we prove the convergence of the learners’ beliefs to the truth and, in the process, explore the relationship between the convergence rate and the degree centrality of the communication graphs.