A Study of Holes: Topological Analysis Reveals Crowd Dynamics Regimes in a Bidirectional Corridor Scenario

Jul 16, 2026

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This study uses topological data analysis to uncover structure in crowd dynamics. Persistent homology, via counts of connected components and holes, characterizes relational patterns in data without relying on expert-designed features. The method is applied to a simulated time series of pedestrian positions in a wide corridor with uni- or bidirectional flow, where proximity defines pairwise connectivity. Persistence signatures are assembled into CROCKERs matrices, whose principal components separate parameter regimes. Results hold under symmetry and support classifying crowd dynamics without assumptions about spatio-temporal patterns.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages5

Reading Time13 min

Abstract

1 Introduction: The topological view of pedestrian dynamics

2 Methods: Obtaining macroscopic observables using persistent homology

arXiv :2607 .06086v2 [math .DS] 8 Jul 2026 A STUDY OF HOLES: TOPOLOGICAL ANALYSIS REVEALS CROWD DYNAMICS REGIMES IN A BIDIRECTIONAL CORRIDOR SCENARIO A PREPRINT Sabrina Desiree Kern and Gerta Köster Faculty of Informatics and Mathematics Hochschule München – University of Applied Sciences Lothstraße 34, 80335 München, Germany [sabrina.kern@hm.edu](sabrina.kern@hm.edu) ABSTRACT This study harnesses topological analysis in an attempt to reveal structure in the dynamics of a crowd. Topology and in particular persistent homology characterizes relational structures in data through the number of connected components and holes, that is, a loop of pairwise connection with no connections across it. We apply this universal data analysis method to a simulated time series of individual pedestrian positions of a crowd moving through a wide corridor – either uni-or bidirectional. We consider two pedestrians to be connected, when they are sufficiently close. This approach leads to two matrices containing the persistence signatures for the whole time series, so-called CROCKERs. Despite the high level of data abstraction, the CROCKERs’ first two principal components on time-delayed positional data show a clear separation of the different parameter configurations. This holds up to symmetry. Our results support our claim that persistent homology is a useful tool to characterize crowd dynamics without introducing any prior assumptions about the detectable spatio-temporal patterns. Keywords dynamical systems · persistent homology · crowd dynamics · bidirectional flow · corridor 1 Introduction: The topological view of pedestrian dynamics When looking at a crowd from above, we see areas of closeness and gaps that seem to depend on the topography, but also on individual and collective destinations, and even the psychological state of the crowd. Relative positions in a crowd of fans moving quickly towards a concert of their favourite artist will certainly differ from those in a crowd leisurely waiting for a train. The occurrence of topological structures, that is, connectivity and holes, inspires us to investigate crowd dynamics through the lens of topology. Topology and in particular persistent homology offers a tool to analyze these geometrical structures without exerting any expert knowledge. We are encouraged by previous work in [1], who successfully classify emergent collective phenomena in interaction models from biology. Interestingly, their approach does not need any hand-crafted metrics to distinguish different collective patters and to solve the inverse problem of uncovering the parameters they used for their simulated time series data. In contrast to works such as [3], we plan to use the pedestrians’ positions directly, without introducing additional macroscopic observables such as the velocity of the crowd. Our aim of this work is to understand when and how collective phenomena in crowds can be identified as structurally different, that is, when and how they can be classified without using hand-crafted metrics. This requires us to transfer established methods from topological data analysis to this new area of application, namely crowd dynamics. As we are not aware of other studies using persistent homology to analyze pedestrian crowds through their positions, this work starts on a simple but dynamically interesting example: a corridor with uni-or bidirectional flow. This minimal example already contains qualitative phenomena such as lane formation while keeping the number of parameters rather small. To better understand how much information is captured in the case of a dynamic crowd, we want to compare several scenarios in a low-dimensional state space, where we expect scenarios with similar spatio-temporal dynamics to cluster. This in turn provides some insight into the capabilities of the topological data analysis in contrast to using hand-crafted metrics. 2 Methods: Obtaining macroscopic observables usi

FAQs

How does the study model pedestrian interactions to compute topology from data?

Pedestrians are considered connected when they are sufficiently close. This proximity rule defines the relational structure used in persistent homology.

What is the role of persistent homology in analyzing crowd dynamics?

Persistent homology characterizes relational structures through the number of connected components and holes (loops of pairwise connection without crossing connections). These features summarize patterns in the positional time series.

How are different corridor scenarios distinguished in the results?

Persistence signatures are organized into CROCKERs matrices, and the first two principal components on time-delayed positional data separate different parameter configurations, with separation maintained up to symmetry.

Jul 16, 2026

Upload by Miles