A64 Group fragmentation

Topic

The swarm remains united thanks to social cohesion maintained by visual, acoustic, or chemical signals exchanged among individuals; swarm fragmentation occurs precisely when this social cohesion is disrupted—either because a physical disturbance (such as wind or an obstacle) disorganizes the swarm, or because the very signals holding it together are lost.

The Vicsek model with decay describes what happens when the swarm is disturbed in this way: the collective velocity at which the swarm moves as a single body begins to decay exponentially—at a rate specific to each fragmentation event—and this decay is the first sign that the swarm is ceasing to behave as a single unit.

When the swarm fragments near a critical percolation point—the same type of threshold separating a connected swarm from a collapsed one during failure cascades—the sizes of the resulting fragments are not evenly distributed but instead follow a power law: a vast number of small fragments appear alongside very few large ones, rather than fragments of roughly similar sizes.

The swarm's susceptibility to fragmenting in this manner depends on an internal balance known as the cohesion radius: on one hand, each individual in the swarm has an inherent tendency to move away from the rest at a certain speed; on the other, there is a swarm escape rate that measures how quickly individuals break free from that collective cohesion. Ultimately, the cohesion radius represents the equilibrium point between these two opposing tendencies.

Right at that critical percolation point, the fracture splitting the swarm into fragments propagates at virtually zero speed—a sign that the swarm is on the very brink of collapse; As the swarm moves away from that critical point—whether toward a more cohesive state or a more fragmented one—the speed at which the fracture propagates increases, following a scaling law characteristic of critical phenomena.

Once the swarm has fragmented, the separated fragments can still reassemble, following Smoluchowski’s theory of random encounters: the time it takes for two fragments to meet again increases with the square of the distance separating them, meaning that fragments left in close proximity rejoin the original swarm much faster than those that ended up far apart.

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