Topological Data Analysis of Biological Aggregation Models

TL;DR

This paper employs topological data analysis to study simulation data from biological aggregation models, revealing structural insights beyond traditional order parameters.

Contribution

It introduces a novel application of persistent homology to analyze agent-based models of biological groups, providing new topological insights into their dynamics.

Findings

Topological features reveal events not captured by order parameters

Betti numbers track structural changes over simulation time

Topological analysis complements traditional measures of aggregation

Abstract

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.