A bio-inspired distributed strategy to infer the size of a network system

Collective animal behavior has served as an invaluable source of inspiration for the design of optimization, estimation, and control algorithms in engineered systems, such as robotic swarms or the electrical power grid. Recent empirical evidence on fish collective behavior indicates that schooling — highly coordinating swimming— is modulated by the number of subjects in a shoal. Motivated by these findings, we conducted an analysis of individual fish swimming in groups. Interestingly, we found that the statistical dispersion of turn rate scales with the number of animals in a group. Inspired by this finding, we develop a simple yet effective algorithm for inferring the group size of a multi-agent system using local information only. In our formulation, each agent updates its state according to a first-order stochastic differential equation and communicates with neighbor agents. Similar to the empirical observations in fish shoals, we show that the statistical dispersion of the resulting emergent probability density function scales with the group size. This enables each agent in a network to only use local information to provide an estimate of the total number of agents. Our theoretical results are illustrated with a set of representative examples demonstrating the effectiveness of our approach.