My latest paper introduces the concept of critical numerosity: A number of individuals above and below which the behavior of a collective qualitatively differs.

Abstract 🔗

A remarkable fact is that natural collectives, despite comprising individuals who may not know their number, can exhibit behavior that depends sensitively on this number. This paper demonstrates that an extreme instance of this phenomenon is possible: Collectives of number-oblivious individuals can have critical numerosities above and below which their behavior qualitatively differs. To do so, we formalize the concept of critical numerosity in terms of a family of zero–one laws and prove that a model of collective motion, called chain activation and transport (CAT), has critical numerosities.

CAT describes the collective motion of $n \geq 2$ individuals as a Markov chain that rearranges $n$-element subsets of the $d$-dimensional grid, $m < n$ elements at a time. According to the individuals’ dynamics, with each step, CAT removes $m$ elements from the set and then progressively adds $m$ elements to the boundary of what remains, in a way that favors the consecutive addition and removal of nearby elements. This paper proves that, if $d \geq 3$, then CAT has a critical numerosity of $n_c = 2m+2$ with respect to the behavior of its diameter. Specifically, if $n < n_c$, then the elements form one “cluster,” the diameter of which has an a.s.–finite limit infimum. However, if $n \geq n_c$, then there is an a.s.–finite time at which the set consists of clusters of between $m+1$ and $2m+1$ elements, and forever after which these clusters grow apart, resulting in unchecked diameter growth.

The existence of critical numerosities means that collectives can exhibit “phase transitions” that are governed purely by their numerosity and not, for example, their density or the strength of their interactions. This fact challenges prevalent beliefs about collective behavior; inspires basic scientific questions about the role of numerosity in the behavior of natural collectives; and suggests new functionality for programmable matter, like robot swarms, smart materials, and synthetic biological systems. More broadly, it demonstrates an opportunity to explore the possible behaviors of natural and engineered collectives through the study of random processes that rearrange finite sets.