  We apply tools from topological data analysis to two mathematical models
inspired by biological aggregations such as bird flocks, fish schools, and
insect swarms. Our data consists of numerical simulation output from the models
of Vicsek and D'Orsogna. These models are dynamical systems describing the
movement of agents who interact via alignment, attraction, and/or repulsion.
Each simulation time frame is a point cloud in position-velocity space. We
analyze the topological structure of these point clouds, interpreting the
persistent homology by calculating the first few Betti numbers. These Betti
numbers count connected components, topological circles, and trapped volumes
present in the data. To interpret our results, we introduce a visualization
that displays Betti numbers over simulation time and topological persistence
scale. We compare our topological results to order parameters typically used to
quantify the global behavior of aggregations, such as polarization and angular
momentum. The topological calculations reveal events and structure not captured
by the order parameters.

Quantitative Methods;Algebraic Topology;Adaptation and Self-Organizing Systems