Mathematicians use 'holes' in data to reveal crowd flow patterns in corridors

Sabrina Desiree Kern and Gerta Köster have introduced a novel method to analyze crowd dynamics using topological data analysis, specifically persistent homology. Their work, presented at Traffic and Granular Flow 2026, treats each pedestrian as a node in a topological space, where two individuals are connected if they are within a certain distance of each other. Over a simulated time series of positions in a wide corridor (both unidirectional and bidirectional flows), they compute the number of connected components and the number of 'holes' — loops of pairwise connections with no internal connections. These persistence signatures are aggregated into matrices called CROCKERs (Cumulative Rank Ordered Crocker plots).

Applying principal component analysis to the CROCKERs reveals distinct clusters corresponding to different crowd configurations (e.g., flow direction and density). The separation holds even when accounting for symmetry in the data. This approach demonstrates that persistent homology can extract meaningful structural patterns from highly abstracted spatial data without making prior assumptions about the spatiotemporal patterns present. The implications reach beyond corridors: the same technique could be used to analyze evacuation dynamics, pedestrian intersections, or even animal swarm behavior, offering a universal tool for characterizing collective motion from raw position data.