Data from: Topological data analysis of biological aggregation models

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.