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Inferring the size of a collective from the motion of a few accessible units is a fundamental problem in network science and interdisciplinary physics. Here, we recognize stochasticity as the commodity traded in the units’ interactions. Drawing inspiration from the work of Einstein-Perrin-Smoluchowski on the discontinuous structure of matter, we use the random motion of one unit to identify the footprint of every other unit. Just as the Avogadro’s number can be determined from the Brownian motion of a suspended particle in a liquid, the size of the collective can be inferred from the random motion of any unit. For self-propelled Vicsek particles, we demonstrate an inverse proportionality between the diffusion coefficient of the heading of any particle and the size of the collective. We provide a rigorous method to infer the size of a collective from measurements of a few units, strengthening the link between physics and collective behavior.

 

Abstract In recent years, three-dimensional (3D) construction printing has emerged as a viable alternative to conventional construction methods. Particularly promising for large scale construction are collective printing systems consisting of multiple mobile 3D printers. However, the design of these systems typically relies on the assumption of continuous communication between the printers, which is unrealistic in dynamically changing construction environments. As a first step toward decentralized collective 3D printing, we explore an active sensing framework allowing individual agents to reconstruct the shape of the structure, toward assessing other agents' progress in the absence of direct communication. In this vein, the shape of the structure is discretized as a 2D lattice embodying its topology, such that the problem is equivalent to the inference of a network. We leverage environmental modifications introduced by each agent through the printing of new layers to track the structure evolution. We demonstrate the validity of a sequential approach based on system identification through numerical simulations. Our work paves the way to decentralized collective 3D construction printing, as well as other applications in collective behavior that rely on the physical medium to transfer information among agents.