Inferring the size of a collective of self-propelled Vicsek particles from the random motion of a single unit

small moving particles with one magnified under glass

BROOKLYN, New York, Monday, April 11, 2022 – Collective dynamics are ubiquitous in the natural world. From neural circuits to animal groups, there are countless instances in which the interactions among large numbers of elementary units bestow surprisingly complex patterns of tantalizing beauty on the collective. One of the longstanding goals of researchers in many fields is to understand behaviors of a large group of individual units by monitoring the actions of a single unit. For example, an ornithologist can learn many things about the behaviors of a flock by monitoring only a single bird.

Of greater difficulty is understanding the size of a collection of units by observing a single unit. No matter how many birds one tags with monitoring equipment, one can never be assured of having tagged the entire flock. Yet, while the ability to calculate the size of a collective from individual behaviors would be a key tool for any field, there are only a handful of recent papers trying to tackle the seemingly unsolvable problem.

In a newly published study appearing in Physics Communication, investigators led by Maurizio Porfiri, Institute Professor of Mechanical and Aerospace Engineering and Biomedical Engineering, and a member of the Center for Urban Science and Progress (CUSP) at the NYU Tandon School of Engineering; and Pietro De Lellis of the University of Naples, Italy, offer a paradigm to solve this problem, one that builds upon precepts that can be traced back to the work of Einstein. 

By observing a system of self-propelled Vicsek particles — a mathematical conceptualization of motion and swarming of particles — as a universal model for collective dynamics, they 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 under particular parameters, such as a known and constant temperature. 

Broadly, the study provides a rigorous, mathematically backed method to infer the size of a realistic collective from measurements of some of its units, whose random motion contains the footprints of the entire system. The theoretical underpinnings of the method provide further evidence for the analogies identified by Einstein between interdisciplinary research in the collective behavior of animal groups and modern physics. Future work in this vein may study real collectives, from insect swarms to bird flocks, fish schools, and human crowds.

The research was supported in part by the National Science Foundation and the University of Naples Federico II and Compagnia di San Paolo, Istituto Banco di Napoli.

The study, “Inferring the size of a collective of self-propelled Vicsek particles from the random motion of a single unit,” can be found at:


About the New York University Tandon School of Engineering

The NYU Tandon School of Engineering dates to 1854, the founding date for both the New York University School of Civil Engineering and Architecture and the Brooklyn Collegiate and Polytechnic Institute. A January 2014 merger created a comprehensive school of education and research in engineering and applied sciences as part of a global university, with close connections to engineering programs at NYU Abu Dhabi and NYU Shanghai. NYU Tandon is rooted in a vibrant tradition of entrepreneurship, intellectual curiosity, and innovative solutions to humanity’s most pressing global challenges. Research at Tandon focuses on vital intersections between communications/IT, cybersecurity, and data science/AI/robotics systems and tools and critical areas of society that they influence, including emerging media, health, sustainability, and urban living. We believe diversity is integral to excellence, and are creating a vibrant, inclusive, and equitable environment for all of our students, faculty and staff. For more information, visit