A recently discovered letter written by Albert Einstein has identified deep, common roots between the interdisciplinary field of animal collective behavior and modern physics. Let the collective be a school of fish, a human crowd, or a flock of birds, the size of a collective is arguably its most fundamental property, but seldom do we have access to this quantity. In most cases, we, the researchers, have access to the time-series of only a few accessible individuals, from which we shall draw inferences about the size of the collective. For example, the size of a group of carrier pigeons – one example that Einstein envisaged would “lead to the understanding of some physical process which is not yet known” – is difficult, if not impossible, to measure with existing remote sensing technology. All we can do is to tag a few birds: is the size of the group predictable from the motion of the tagged individuals?

Identifying the size of a collective

Only a handful of recent studies have attempted to devise methodologies for estimating the size of a group from measurements of some of its units. The recent breakthrough by Haene et al. indicates an important link between the rank of a detection matrix constructed from transient dynamics and the size of the collective. A recent study from Porfiri has offered further mathematical backing to this method, identifying a control-theoretic basis in the concept of observability. Alternative approaches have been proposed by Tang et al. and Tyloo and Delabay, exploiting transient and steady-state dynamics. The Achille’s heel of all these existing methods is the assumption of deterministic dynamics, which, in turn, limits their use to an idealized, noiseless setting – a far distant case from the reality of collective animal behavior.

Our paper

In this article, we take a radically different approach by treating noise as a commodity to reconstruct the size of a collective from the random motion of any of its units. Following a conceptual logical circle that goes back to the seminal work of Einstein, Perrin, and Smoluchowski on the estimation of the Avogadro’s number from Brownian motion, we demonstrate that knowledge of the mean square dynamics of a single unit is sufficient to determine the size of the entire collective. Using the words of Feynman regarding the experiments of Perrin, “one of the earliest determinations of the number of atoms was by the determination of how far a dirty little particle would move if we watched it patiently under a microscope for a certain length of time.” 

In our paper, we show that the mean square dynamics of a single unit is sufficient to infer the size of a collective. We demonstrate our approach on a system of self-propelled Vicsek particles as a universal model for collective dynamics, for which we show that the time rate of growth of the mean square heading of any particle is sufficient to predict the number of particles in the system. We explain this concept by drawing an analogy between the Vicsek model and the classical kinetic theory of gases, where the mean square displacement of any particle is given by λ ∝ Dt, with t being time and D the diffusion coefficient. The latter can be estimated as D ∝ k_BTV/(mND²), where k_B is Boltzmann's constant, T the absolute temperature, V the volume of the gas, d the particle diameter, m the particle mass, and N the number of particles. One can estimate the number of particles in the gas from the rate of growth of the mean square displacement, upon some knowledge of the thermodynamic state of the gas and the properties of its particles.

Pursuing this analogy, we started from the observation that in groups of Vicsek particles, the emergence of collective behavior is caused by some degree of connectivity between the particles. The persistence of an average connectivity implies that the motion of one unit will contain a significant footprint of the size of the entire system. We numerically and mathematically demonstrate that the inference of the size of the system from the motion of a single particle only requires knowledge about the variance of the added noise, which, indeed, could be viewed as the thermodynamic analog of the temperature for collective dynamics. Different from the seminal experiments of Perrin, our focal unit is not a macroscopic particle in a bath of microscopic particles, rather, it is one of the very particles comprising the collective whose trajectory is accessible by the experiment, that is, a tagged pigeon in the flock.

Looking ahead

The theoretical findings of this study offer a compelling base to tackle the identification of the size of real collectives, from insect swarms to bird flocks, fish schools, and human crowds. Albeit we demonstrated the approach on a popular and general model of collective behavior, we expect that scaling to real collectives will pose intriguing challenges for the research community, toward capturing intricacies and complexities of collective behavior. For example, how can we estimate the thermodynamic temperature of the collective from data? Perhaps a first step in this direction should seek to apply the approach to more detailed models of collective behavior, including, for instance, attraction and repulsion rules. Following the path traced by Einstein in his 1949 letter, we expect that many other salient features of real collectives, beyond their size, can be better understood and practically estimated by building analogies with fundamental principles in physics. For instance, an open question in the study of collective behavior pertains to understanding which units are the most vulnerable to external perturbation. Interesting physical analogies have already been pointed out in the context of collective human behavior, with preliminary research pointing at the potential of environmental or economic shocks to trigger cascading migrations at a global, worldwide scale.