Topological data analysis of collective dynamics

Chad Topaz
Department of Mathematics and Statistics, Williams College, Williamstown

From nanoparticle assembly to synchronized neurons to locust swarms, collective behaviors abound anywhere in nature that objects or agents interact. Investigators modeling collective behavior face a variety of challenges involving data from simulation and/or experiment. These challenges include exploring large, complex data sets to understand and characterize the system, inferring the model parameters that most accurately reflect a given data set, and assessing the goodness-of-fit between experimental data sets and proposed models. Topological data analysis provides a lens through which these challenges may be addressed. This talk consists of three parts. In the first part, I apply topological data analysis to the seminal aggregation model of Vicsek et al. (1995) in order to identify dynamical events that traditional methods do not. In the second part, I use topological data analysis to choose between unbiased correlated random walk models that potentially describe motion tracking experiments on pea aphids. Finally, moving towards a theory of reduced topological descriptions of complex behavior, I present open questions on the topology of random data, complementing research in random geometric graph theory. Throughout the talk, the key approach is to characterize a system's dynamics via the time-evolution of topological invariants called Betti numbers, accounting for persistence of topological features across multiple scales. No prior knowledge of topology is necessary.