Efficient flocking: metric versus topological interactions.

Vijay Kumar,Rumi De

doi:10.1098/rsos.202158

Abstract

Flocking is a fascinating phenomenon observed across a wide range of living organisms. We investigate, based on a simple self-propelled particle model, how the emergence of ordered motion in a collectively moving group is influenced by the local rules of interactions among the individuals, namely, metric versus topological interactions as debated in the current literature. In the case of the metric ruling, the individuals interact with the neighbours within a certain metric distance; by contrast, in the topological ruling, interaction is confined within a number of fixed nearest neighbours. Here, we explore how the range of interaction versus the number of fixed interacting neighbours affects the dynamics of flocking in an unbounded space, as observed in natural scenarios. Our study reveals the existence of a certain threshold value of the interaction radius in the case of metric ruling and a threshold number of interacting neighbours for the topological ruling to reach an ordered state. Interestingly, our analysis shows that topological interaction is more effective in bringing the order in the group, as observed in field studies. We further compare how the nature of the interactions affects the dynamics for various sizes and speeds of the flock.

Highlights

Cohesive group formation is one of the most eye-catching displays in nature
We study the dynamics of flocking by varying the range of the interaction radius, R, in the case of metric and the number of interacting neighbours, Nr, for the topological interactions
We have investigated the dynamics for a wide range of parameter values; here, we present the results for some representative values by keeping the initial box size, L = 25, αs = 1, and the flock size, N = 100, 200, 300, 500

Summary

Introduction

Cohesive group formation is one of the most eye-catching displays in nature. It is observed among various species, such as, flock of birds [1,2], school of fishes [3], swarm of prey [4,5], colony of bacteria [6], aggregation of cells [7] and pedestrian crowd [8]. Using a simple model with metric rules, Couzin et al have shown how efficient information transfer and decision-making can occur in animal groups [24] Another self-propelled particle model introduced by Bhattacharya and Vicsek indicates the mechanisms of the synchronized landing of a flock of birds performing foraging flights [25]. Using network and graph-theoretic approaches coupled with a dynamical model, Shang and Bouffanais have studied the consensus reaching process with topologically interacting self-propelled particles They have shown regardless of the group size, a value of close to 10 neighbours speeds up the rate of convergence to the consensus to an optimal level where all particles interact with the entire group [31]. The topological interaction turns out to be beneficial and effective in bringing order in the flock

Theoretical model

Results

Metric interaction: influence on flocking state

Topological interaction: influence on flocking state

Metric versus topological interactions

Discussion