A study of holes: Topological analysis reveals crowd dynamics regimes in a bidirectional corridor scenario

We introduce a data-analysis framework based on filtrations of finite topological spaces. Starting from a finite metric data set, we construct a sequence of coarsening topologies on the same set of points. These topologies give persistence modules and barcodes in the usual way, but they also retain information that is lost when the filtration is reduced to homology. At each level one can examine, for example, which points are topologically indistinguishable, how their minimal neighbourhoods overlap, how connected components merge, and how these features change from one level to the next. We develop the basic theory of these filtrations, establish stability results under suitable hypotheses, and give practical constructions starting directly from a distance matrix. We then study what can be learned from the resulting finite topologies. On synthetic data with known clusters of different shapes, sizes, and densities, we examine how these regions appear among the finite-topological structures and how they merge as the topology coarsens. We also study what happens when points that become uncovered early in the construction are removed and the analysis is repeated. For one-dimensional homology, we use paths in the finite-topological structure to locate cycles and to examine how their appearance is related to the geometry of the data. We finally apply these ideas to two real data sets with quite different structures. On the Paul15 single-cell data, we use the evolving finite topology to examine fine cellular states, their overlaps and relations, their assembly into larger groups, and the effect of removing points that connect these structures. On COIL20, where images of an object are sampled through a full rotation, we study how the cyclic organization of the images is reflected in the finite-topological evolution and in the associated one-dimensional homology.

We propose persistent discrete homology as a tool for topological data analysis and discuss its advantages over the existing methods. In particular, we provide empirical evidence that persistent discrete homology is more noise-resistant than persistent homology of the Vietoris-Rips complex for data coming from non-metric settings.

Crowd movement simulation is essential for pedestrian safety management and facility layout optimization. Data-driven models enhance trajectory prediction accuracy under Euclidean metrics, yet they suffer from excessively high collision rates, especially in bidirectional and multidirectional flows. In this paper, we establish a novel data-driven crowd simulation model that incorporates the pedestrian collision mechanism into the loss function to reduce collisions. A new lateral-acceleration-based collision loss function and a Voronoi-based motion feature extraction approach are proposed. The model is based on a Generative Adversarial Network (GAN) architecture and is termed CPGAN (Collision-Penalized GAN). We evaluate CPGAN in bidirectional flow scenarios, which involve frequent collision avoidance behaviors. Results show that the proposed lateral-acceleration-based collision loss significantly reduces opposite-direction pedestrian collision rates to levels comparable with controlled experiments. CPGAN effectively simulates bidirectional flow, reproducing lane formation and N-t curves. The research outcomes can provide inspiration for integrating pedestrian dynamics mechanisms into loss functions in data-driven crowd simulation.