Coarse-grained variables for particle-based models: diffusion maps and animal swarming simulations

Asmicroscopic (e.g. atomistic, stochastic, agentbased, particle-based) simulations become increasingly prevalent in the modeling of complex systems, so does the need to systematically coarse-grain the information they provide. Before even starting to formulate relevant coarsegrained equations, we need to determine the right macroscopic observables—the right variables in terms of which emergent behavior will be described. This paper illustrates the use of data mining (and, in particular, diffusion maps, a nonlinear manifold learning technique) in coarsegraining the dynamics of a particle-based model of animal swarming. Our computational data-driven coarse-graining approach extracts two coarse (collective) variables from the P. Liu · H. R. Safford Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08544, USA e-mail: [email protected] Present address: P. Liu Department of Molecular, Cellular and Developmental Biology, Yale University, West Haven, CT 06516, USA Present address: H. R. Safford Woodrow Wilson School, Princeton University, Princeton, NJ 08544, USA e-mail: [email protected] I. D. Couzin Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08540, USA e-mail: [email protected] I. G. Kevrekidis (B) Department of Chemical and Biological Engineering, Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA e-mail: [email protected] detailed particle-based simulations, and helps formulate a low-dimensional stochastic differential equation in terms of these two collective variables; this allows the efficient quantification of the interplay of “informed” and “naive” individuals in the collective swarm dynamics. We also present a brief exploration of swarm breakup and use data-mining in an attempt to identify useful predictors for it. In our discussion of the scope and limitations of the approach we focus on the key step of selecting an informative metric, allowing us to usefully compare different particle swarm configurations.