Adaptive Cucker-Smale Networks: Limiting Laplacian Time-Varying Dynamics

TL;DR

This paper introduces a new framework for modeling opinion formation and clustering using adaptive Laplacian dynamics on time-varying networks, extending classical consensus models with co-evolutionary features.

Contribution

It develops a mathematical analysis of opinion dynamics on adaptive networks by examining the singular limit of fast adaptation, providing a general methodology for linear consensus models.

Findings

Analysis of asymptotic behavior of Laplacian dynamics on temporal graphs

Explanation of opinion clustering phenomena in adaptive networks

Methodology applicable to various classes of time-varying networks

Abstract

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.