Effect of Spatial Percolation on the Convergence of a Graph Coloring Boid Swarm

Abstract

<jats:p>Graph Coloring Boid Swarm (GCBS) is the successful application of Reynolds' Boid Swarm to solve the graph coloring problem, following appropriate mapping of the problem into boids' behaviors and interpretation of the global swarm states as problem solutions. A population P of boids moves in a closed torus-shaped space S. Each individual boid perceives a disk of radius R of its surrounding space, being able to exchange information with other boids lying inside this area of perception. Two mutually perceiving boids exchanging information are connected. In this paper we show that the ratio of the radius of perception to the space size can be critical for the convergence of the Boid Swarm to the optimal configurations. First, we derive the Percolation threshold for Elementary Boid Swarms (EBS), whose dynamics are the aggregation of the boids into spatial clusters. When the radius of perception is above this percolation threshold for Elementary Boid Swarms (EBS), whose dynamics are the aggregation of the boids into spatial clusters. When the radius of perception is above this percolation threshold there is complete connectivity and convergence to a single cluster. Second, we show empirically the existence of such a Percolation effect on the Graph Coloring Boid Swarm (GCBS). We find that complete connectivity has an adverse effect on the performance of the GCBS. Finally we show a comparison of GCBS with some other algorithms over a family of generated graphs.</jats:p>