2020/05/27 by Jan Haškovec, Haskovec, Jan · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #34D05 #34K05 #82C22 #92D50 #Complex Network Analysis Techniques #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Opinion Dynamics and Social Influence #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2005.13589
openalex publication_date 2020/05/27 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We present a simple proof of asymptotic consensus in the discrete\nHegselmann-Krause model and flocking in the discrete Cucker-Smale model with\nrenormalization and variable delay. It is based on convexity of the\nrenormalized communication weights and a Gronwall-Halanay-type inequality. The\nmain advantage of our method, compared to previous approaches to the delay\nHegselmann-Krause model, is that it does not require any restriction on the\nmaximal time delay, or the initial data, or decay rate of the influence\nfunction. From this point of view the result is optimal. For the Cucker-Smale\nmodel it provides an analogous result in the regime of unconditonal flocking\nwith sufficiently slowly decaying communication rate, but still without any\nrestriction on the length of the maximal time delay. Moreover, we demonstrate\nthat the method can be easily extended to the mean-field limits of both the\nHegselmann-Krause and Cucker-Smale systems, using appropriate stability results\non the measure-valued solutions.\n