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We develop a topological analysis of robust traffic pace patterns using persistent homology. We develop Rips filtrations, parametrized by pace, for a symmetrization of traffic pace along the (naturally) directed edges in a road network. Our symmetrization is inspired by recent work of Turner (2019Algebr. Geom. Topol.191135–1170). Our goal is to construct barcodes which help identify meaningful pace structures, namely connected components or ‘rings’. We develop a case study of our methods using datasets of Manhattan and Chengdu traffic speeds. In order to cope with the computational complexity of these large datasets, we develop an auxiliary application of the directed Louvain neighborhood-finding algorithm. We implement this as a preprocessing step prior to our main persistent homology analysis in order to coarse-grain small topological structures. We finally compute persistence barcodes on these neighborhoods. The persistence barcodes have a metric structure which allows us to both qualitatively and quantitatively compare traffic networks. As an example of the results, we find robust connected pace structures near Midtown bridges connecting Manhattan to the mainland.

 

Abstract This article proposes several advances to sparse nonnegative matrix factorization (SNMF) as a way to identify large-scale patterns in urban traffic data. The input to our model is traffic counts organized by time and location. Nonnegative matrix factorization additively decomposes this information, organized as a matrix, into a linear sum of temporal signatures. Penalty terms encourage this factorization to concentrate on only a few temporal signatures, with weights which are not too large. Our interest here is to quantify and compare the regularity of traffic behavior, particularly across different broad temporal windows. In addition to the rank and error, we adapt a measure introduced by Hoyer to quantify sparsity in the representation. Combining these, we construct several curves which quantify error as a function of rank (the number of possible signatures) and sparsity; as rank goes up and sparsity goes down, the approximation can be better and the error should decreases. Plots of several such curves corresponding to different time windows leads to a way to compare disorder/order at different time scalewindows. In this paper, we apply our algorithms and procedures to study a taxi traffic dataset from New York City. In this dataset, we find weekly periodicity in the signatures, which allows us an extra framework for identifying outliers as significant deviations from weekly medians. We then apply our seasonal disorder analysis to the New York City traffic data and seasonal (spring, summer, winter, fall) time windows. We do find seasonal differences in traffic order. 

The behavior of the optimal velocity model is investigated in this paper. Both deterministic and stochastic perturbations are considered in the Optimal velocity model and the behavior of the dynamical systems and their convergence to their associated averaged problems is studied in detail. 

Traffic flow models have been the subject of extensive studies for decades. The interest in these models is both as the result of their important applications as well as their complex behavior which makes them theoretically challenging. In this paper, an optimal velocity dynamical model is considered and analyzed.We consider a dynamical model in the presence of perturbation and show that not only such a perturbed system converges to an averaged problem, but also we can show its order of convergence. Such understanding is important from different aspects, and in particular, it shows how well we can approximate a perturbed system with its associated averaged problem. 

This article proposes a data-driven combination of travel times, distance, and collision counts in urban mobility datasets, with the goal of quantifying how intertwined traffic accidents are in the road network of a city. We devise a bi-attribute routing problem to capture the tradeoff between travel time and accidents. We apply this to a dataset from New York City. By visualizing the results of this computation in a normalized way, we provide a comparative tool for studies of urban traffic.