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Topological Data Analysis of Biological Aggregation Models
Chad M. Topaz, Lori Ziegelmeier, Tom Halverson
TL;DR
The paper asks whether topology can complement classical order parameters in characterizing collective behavior in biological aggregation models. It applies persistent homology to Vicsek and D’Orsogna simulations and visualizes Betti numbers across scale and time. The analysis distinguishes behaviors and detects events or structures that order-parameter time series do not capture.
Problem
The paper examines whether topology offers a natural complement to classical order parameters for characterizing collective behavior in biological aggregations.
Method
The authors compute persistent homology and the first few Betti numbers of simulation point clouds from the Vicsek and D’Orsogna models, visualizing them with CROCKER across proximity scale and simulation time.
Results
Topological calculations distinguish Vicsek simulations that the alignment order parameter cannot, find topological similarity across different order-parameter time series, and recognize a double mill in D’Orsogna’s model.
Takeaways & Limitations
Topological data analysis provides an additional technology for understanding aggregation group behavior and can reveal dynamical events not captured by classical order parameters.
Takeaways & Limitations
The study usually calculates only b0 and b1 because higher Betti numbers are computationally costly, and persistence over simulation time lacks the filtration guarantee available across scale.
Abstract
from arXiv · showhide
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.
Introduction
Biological aggregations are studied through models, field data, and global order parameters, but this paper investigates topology as an additional way to characterize collective behavior. It applies persistent homology and CROCKER visualizations to simulation data to reveal dynamical structure beyond classical metrics.
- Biological aggregations include fish schools, bird flocks, insect swarms, and mammal herds whose members interact through attraction, repulsion, and alignment.
- Field studies use organism positions and sometimes velocities to infer individual motion rules or characterize collective dynamics.
- Classical analyses calculate global metrics such as polarization, angular momentum, neighbor counts, and nearest-neighbor distance from positions and velocities.
- Polarization and angular momentum distinguish several behavioral regimes, while absolute angular momentum is needed to distinguish single and double mills.
- The paper uses persistent homology on simulated aggregation data and introduces CROCKER to display Betti numbers across proximity scale and simulation time.
- The paper distinguishes its use of topology from the aggregation literature’s use of topology to mean agents’ coupling scheme.
Topological Data Analysis and Persistent Homology
Persistent homology builds simplicial complexes from point clouds across proximity scales, then tracks homological holes and their lifetimes. Betti numbers summarize these features, while barcodes visualize their persistence without requiring a priori selection of one scale.
- Homology quantifies connected components, topological circles, trapped volumes, and other holes in a topological space sampled by finite data points.
- Forming a Simplicial Complex: A simplicial complex assembles vertices, edges, triangles, tetrahedra, and higher-dimensional simplices while retaining every nonempty face of each simplex.
- Forming a Simplicial Complex: The Vietoris-Rips complex includes every collection of k+1 points whose pairwise distances are less than the proximity parameter ε.
- Forming a Simplicial Complex: The Rips complex is used because it is more computationally tractable than the Čech complex.
- Homology: Homology identifies cycles modulo boundaries, so cycles differing by a boundary represent the same homology class.
- Homology: The kth Betti number is the dimension of kth homology and counts independent k-dimensional holes, including components for b0, circles for b1, and trapped volumes for b2.
- Persistence: Because Rips homology depends strongly on ε, barcodes record when homology classes form and disappear across proximity scales.
Computational Methods and Data Visualization
The study computes persistent homology from simulation point clouds and introduces CROCKER contour diagrams to track Betti numbers across proximity scale and simulation time.
- Simulation data: Vicsek simulations represent each agent by two-dimensional position and angular heading in a square domain with periodic boundary conditions.The heading is defined on a bounded angular interval, and simulations are performed in Matlab.
- Computational considerations: The computational cost depends on the number of simplices, with Gaussian elimination over a finite field taking worst-case O(m^3) for m k-simplices.The number of k-simplices depends on the proximity parameter ε and the point-cloud configuration, with an upper bound of O(n^(k+1)).
- Point-cloud processing: Each simulation time step is treated as a static point cloud in position-heading or position-velocity space for topological analysis.The R implementation uses the phom package to construct Rips complexes and calculate topological barcodes.
- Persistent homology: The analysis calculates b0(ε) and b1(ε) over closely spaced proximity parameters ε, capturing connected components and one-dimensional holes.The Rips complex is recomputed across discrete values of ε.
- CROCKER visualization: CROCKER displays bk(ε, t) as a contour diagram, with simulation time on the horizontal axis and proximity parameter ε on the vertical axis.It subsamples time steps while retaining several hundred snapshots per simulation and plots distinct contours only for bk(ε, t) < 5.
- CROCKER visualization: Large contour regions represent topological features persisting across proximity scale and simulation time, while Betti numbers remain nonnegative integers.Values bk(ε, t) ≥ 5 are grouped together rather than shown with separate contours.
Analysis of the Vicsek Model
The Vicsek model produces distinct collective states whose topological signatures reveal clustering, alignment, and spatial coverage beyond the information provided by the order parameter. Persistent Betti numbers track these structures across simulation time and proximity scale.
- Model and initial data: The Vicsek model updates each particle’s heading from nearby neighbors’ average headings plus noise, then moves it a fixed distance on a periodic square.The model uses position and velocity, with heading represented by the velocity angle.
- Model and initial data: The random initial point cloud occupies a three-torus T3, whose first Betti numbers are b = (1, 3, 3, 1).The three persistent b1 bars represent the torus’s three topological circles, while three b2 bars represent its trapped volumes.
- Global behaviors: The model exhibits clusters moving in different directions, correlated random motion, or highly polarized motion, depending on density and noise.The topological analysis is applied alongside the normalized average velocity order parameter ϕ(t).
- Simulation #1: At t = 2080 and t = 2150, topology distinguishes different cluster-merging scales that are not visible in snapshots or reflected by ϕ(t).At t = 2080, four, three, two, and one components appear across successive ε ranges; at t = 2150, the transition from four to two occurs over a much smaller ε range.
- Simulation #1: In simulation #1, b0 reveals persistent cluster formation and merging or fragmentation, while b1 shows rapid loss of the initial three circles followed by one persistent circle.The remaining circle is consistent with coverage across one spatial dimension after alignment and loss of coverage in the other dimensions.
Analysis of the D’Orsogna Model
The D’Orsogna simulations develop mill structures whose topology changes clearly over time and proximity scale. Betti-number contours identify a double mill through two persistent circles and two connected components.
- Model and mill structures: The attractive-repulsive D’Orsogna model produces rotating rings, traveling swarms, and vortex states called mills.The model describes self-propelled particles interacting through attraction and repulsion.
- Model and mill structures: Mills are annular formations around a hollow core; double mills contain particles rotating in both clockwise and counterclockwise directions.A single mill has one orientation, whereas a double mill has two.
- Simulation: The N = 500 simulation evolves from a disorganized disk to a hollow-core structure and then a double mill.Snapshots are shown at t = 5, t = 23, and t = 34, with clockwise and counterclockwise particles distinguished by color.
- Comparison with order parameters: Polarization remains low during circular motion, while angular momentum quantifies rotation and equals 1 for a perfect mill.The topological analysis complements these order parameters by resolving the double-mill structure.
- Topological evolution: At times below t ≈20, there is little topological structure; afterward, one or intermittently two connected components appear.At larger scales, b0(ε, t) = 1 indicates one connected component, while smaller-scale components are short-lived and noisy.
- Topological evolution: Two circles at smaller ε and one at larger ε persist for long simulation periods, yielding b = (2, 2), consistent with a double mill.The two mills merge at approximately the proximity scale corresponding to their separation, while the remaining circle disappears at sufficiently large ε.
Conclusions
The paper demonstrates persistent homology as an additional tool for analyzing simulated biological aggregations. CROCKER visualizations reveal structures that standard order parameters miss, while the study remains limited in topological dimension and simulation scope.
- Conclusions: The study proposes topological data analysis as an additional technology for understanding the collective behavior of biological aggregations.This complements established order parameters such as polarization and angular momentum.
- Conclusions: The authors compute persistent homology and Betti numbers for time-evolving point clouds generated by the Vicsek and D’Orsogna models.The Betti numbers count connected components, topological circles, trapped volumes, and related features.
- Conclusions: CROCKER tracks Betti numbers across proximity scale and simulation time, displaying persistent features as large contour-plot regions.It is introduced to interpret homological persistence in evolving aggregation data.
- Conclusions: Topological measures distinguish Vicsek simulations that the usual alignment order parameter cannot, and recognize a double mill in the D’Orsogna model.The authors report that these structures would not have been found by eye before topological detection.
- Limitations and future work: The study calculates only b0 and b1 except in a small number of isolated cases involving b2.Higher Betti numbers could provide additional information but are computationally costly.
- Limitations and future work: The examples use two convenient models and a small number of simulations; larger randomly seeded ensembles could quantify transition timescales and persistence scales more precisely.The authors also identify vineyards, multidimensional persistence, and braids as possible tools for future work.