Quantifying patterns in the Vicsek Model with topological tools

Abstract

In this work, I explore the topological features of aggregation patterns in the Vicsek Model, a widely used framework for describing the collective dynamics of active matter. By varying the three key parameters—population size N, interaction radius R, and noise η, different point sets of self- organising agents are generated. To analyse the emergent structures, I employ topological tools, namely the Euler characteristic and Betti numbers, in both spatial and temporal domains. The Euler characteristic, a fundamental topological invariant, provides insights into system connectivity, while Betti numbers characterise features such as connected components, loops, and voids. Three-dimensional Euler Characteristic Surfaces (ECS) are constructed that carry the summary of the spatio-temporal evolution of the Euler Characteristic. Further, a metric distance, which we name the Euler Metric (EM), is estimated between these surfaces to investigate how system parameters influence aggregation dynamics. Additionally, I analyse order parameters to distinguish between ordered and chaotic regimes, further contextualising the topological findings.