Flocking Time and Graph Volume in Nonlinear Cucker-Smale Models with Dynamic Networks

This study proposes a Cucker-Smale model incorporating dynamic networks and the discrete p-Laplacian ( ), and analyzes the relationship between flocking time-constraints and graph connectivity from both theoretical and experimental perspectives.We establish that this model achieves flocking within a finite time and derive a quantitative relationship between the lower bound of the flocking time and graph volume.Results show that imposing sufficiently small flocking time increases the network volume, and that a larger initial velocity variance requires greater volume under the same time constraint.In our experiments, we trained a transformer-based proximal policy optimization policy to select network structures at each time step.Under the most stringent time constraint, the maximum average number of connections increased sharply, confirming a threshold effect.The negative correlation between the convergence time and average number of connections reflects the theoretical results.This study provides quantitative evidence of the phenomenon that time constraints induce connection mobilization.