A61 Swarm failure cascades

Topic

A swarm can collapse completely due to the loss of just a small number of its individuals; this is known as a failure cascade—an event where an initial loss, small in itself, triggers a chain reaction that ultimately brings down the entire swarm.

The Watts cascade model explains how this chain reaction is triggered within the swarm: each individual has its own failure threshold and fails when the fraction of its neighbors that have already failed exceeds that personal threshold. Consequently, the failure of a few individuals can push their direct neighbors to fail, and those neighbors, in turn, push theirs.

How far this chain reaction spreads depends on the swarm's topological connectivity—that is, how each individual is connected to the others. There is a critical connection probability that separates a regime where the swarm remains stable from one where it collapses completely.

Near this critical probability, the swarm behaves in a particularly sensitive manner, as described by percolation theory: the size of the failure cascade does not merely grow in proportion to the swarm's proximity to that critical probability; instead, it grows increasingly disproportionately as the swarm approaches that critical point, becoming virtually limitless right at the threshold. This explains why the loss of just a few individuals can, under the right conditions, trigger the collapse of an entire swarm. Beyond this description in terms of networks and probability, the same failure cascade can also be described in the physical space occupied by the swarm: a diffusion-advection model describes how the localized loss of a subgroup of individuals propagates spatially through the swarm, moving from one region to the next rather than jumping randomly between distant individuals.

The swarm's ultimate resilience to this risk depends on how connectivity is distributed among its individuals: a swarm where most individuals have similar connectivity withstands both the random loss of individuals and targeted attacks reasonably well. However, a swarm with a highly unequal connectivity distribution—where a few individuals act as hubs that are far more connected than the rest—is particularly vulnerable if the attack targets precisely those central individuals, even though that same swarm might tolerate the random loss of less connected individuals quite well.

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