Adaptive Cucker-Smale networks: Limiting Laplacian time-varying dynamics

Differences in opinion can be seen as distances between individuals, and such differences do not always vanish over time. In this paper, we propose a modeling framework that captures the formation of opinion clusters, based on extensions of the Cucker-Smale and Hegselmann-Krause models to a combined adaptive (or co-evolutionary) network. Reducing our model to a singular limit of fast adaptation, we mathematically analyze the asymptotic behavior of the resulting Laplacian dynamics over various classes of temporal graphs and use these results to explain the behavior of the original proposed adaptive model for fast adaptation. In particular, our approach provides a general methodology for analyzing linear consensus models over time-varying networks that naturally arise as singular limits in many adaptive network models.