# Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph

## TL;DR

This paper investigates how time-delay and velocity alignment influence flocking behavior in the Cucker-Smale model on directed graphs with spanning trees, providing conditions for exponential flocking and threshold phenomena based on decay rates.

## Contribution

The paper introduces new frameworks for analyzing mono-cluster flocking in continuous and discrete Cucker-Smale models with time-delay on directed networks, including explicit decay weight conditions.

## Key findings

- Exponential flocking estimates are established for both continuous and discrete models.
- Threshold phenomena depend on decay rate and graph depth for algebraically decaying weights.
- Numerical examples confirm analytical predictions.

## Abstract

We study dynamic interplay between time-delay and velocity alignment in the ensemble of Cucker-Smale (C-S) particles(or agents) on time-varying networks which are modeled by digraphs containing spanning trees. Time-delayed dynamical systems often appear in mathematical models from biology and control theory, and they have been extensively investigated in literature. In this paper, we provide sufficient frameworks for the mono-cluster flocking to the continuous and discrete C-S models, which are formulated in terms of system parameters and initial data. In our proposed frameworks, we show that the continuous and discrete C-S models exhibit exponential flocking estimates. For the explicit C-S communication weights which decay algebraically, our results exhibit threshold phenomena depending on the decay rate and depth of digraph. We also provide several numerical examples and compare them with our analytical results.