Characteristics of swarms on the edge of fragmentation

Fragmentation of particle swarms into isolated subgroups occurs when interaction forces are weak or restricted. In the restricted case, the swarm experiences the onset of bottlenecks in the graph of interactions that can lead to the fragmentation of the system into subgroups. This work investigates the characteristics of such bottlenecks when the number of particles in the swarm increases. It is shown that certain characteristics of the bottleneck can be captured by considering only the number of particles in the swarm. Considering the case of a connected communication graph constructed in the hypothesis that each particle is influenced by a fixed number of neighboring particles, a limit case is determined for which a lower limit to the Cheeger constant can be derived analytically without the need for extensive algebraic calculations. Results show that as the number of particles increases, the Cheeger constant decreases. Although ensuring a minimum number of interactions per particle is sufficient, in theory, to ensure cohesion, the swarm may face fragmentation as more particles are added to the swarm.

Figure 1

The illustration shows how for $k=\langle N/2\rangle$ the swarm must be connected. Each of the particles in the subgroup on the left must complete its four connections by joining the group on the right.

Figure 2

Particle swarm systems in a two-dimensional space relaxing to different shapes as the number of particles $N$ increases while the number of connections allowed per particle, $k$, is held constant at 60.

Figure 3

Dumbbell emergence in a three-dimensional case. (a) A 130-particle swarm in a dumbbell shape due to the number of connections per particle being limited to 65 and (b) the corresponding adjacency matrix obtained associating each node to one particle, after having them sorted from one end to the other of the dumbbell. Dots represent nonzero entries.

Figure 4

Adjacency matrices for idealized connected (a) and spontaneously relaxed (b) 60 particle swarm.

Figure 5

Comparison of Cheeger constant obtained by the analytic expression simply based on the number of particles and that obtained through numerical simulations averaged over 100 runs with random initial conditions.