From Motifs to Lévy Flights: Modeling Urban Mobility in Bogotá's Public Transport System

TL;DR

The paper addresses how to characterize urban mobility in Bogotá using two years of multimodal smart-card validations by reconstructing recurrent trajectory motifs and building an integrated origin–destination matrix across 2,828 zones. It shows that transition probabilities between zones follow a distance-dependent pattern well described by a Lévy-flight framework, validated through Monte Carlo simulations that reproduce the empirical distance distribution with γ ≈ 1. This approach relies solely on entry records to construct the OD representation and demonstrates temporal stability of mobility structures despite infrastructure changes and fare policy updates. The work provides a data-efficient, generalizable framework for analyzing complex urban transport systems and offers a basis for cross-city comparisons and planning across multimodal networks.

Abstract

In this paper, we study two years of access card validation records from Bogotá's multimodal public transport system, comprising over 2.3 billion trips across bus rapid transit, feeder buses, dual-service buses, and an aerial cable network. By reconstructing user trajectories as motifs, we identify recurrent mobility patterns that extend beyond simple round trips, enabling the construction of an integrated origin-destination (OD) matrix covering 2,828 urban zones. Similarity analysis using the Jensen-Shannon divergence confirms the temporal stability of mobility structures across semesters, despite infrastructure changes and fare policy adjustments. From the obtained OD matrices, we derive transition probabilities between zones and uncover a robust power-law relationship with geographical distance, consistent with Lévy flight dynamics. We validate our model using Monte Carlo simulations showing that reproduces both local and long-range displacements, with similar scaling exponents across time. These findings demonstrate that Bogotá's public transport mobility can be effectively modeled through Lévy processes, providing a novel framework for analyzing complex transportation systems based solely on user access records.

Figures (6)

• Figure 1: Analysis of mobility patterns of public transport users in Bogotá. (a) Frequency of use throughout the day in five-minute intervals. The black line represents the average, and the shaded region indicates the standard deviation of the data for weekdays, Saturdays, and Sundays/Holidays, highlighting the strong regularity of the city, particularly on weekdays. (b) Heatmap showing the spatial distribution of trips during morning hours and (c) afternoon hours. Panels (b)-(c) show the city divided into 2,828 zones of 300 × 300 meters, each containing bus stops and stations across transport modes. The color bar, presented on a logarithmic scale, indicates the number of users, highlighting the urban structure and the predominance of the BRT system over feeder buses.
• Figure 2: Statistical representation of the 11 most frequent motifs among system users. These motifs account for 96.06% of all reconstructed trajectories, which correspond to chains of two to six trips with successive accesses at locations denoted $A, B, \ldots, F$. Each number identifies the diagram of its respective motif, constructed under the assumption that each validation represents the immediate destination of the previous trip; therefore, the final destination remains undefined. The most recurrent pattern corresponds to linear trajectories across different areas without repetitions, highlighting the need to further investigate mobility structures more complex than simple round trips.
• Figure 3: Matrix representation of public transport mobility in Bogotá. (a) OD matrix representing the movement of public transport users across 2,828 zones, corresponding to the 300 m × 300 m partition shown in Fig. \ref{['Fig_1']}. The matrix includes trajectories of two to six trips, which account for 85.04% of the data. The OD matrix was constructed following the motif logic, where the destination corresponds to the immediate subsequent trip. (b) Distance matrix of Bogotá, where values, expressed in meters, represent the geographical distance between origin and destination zones, as indicated by the color bar.
• Figure 4: Comparative study of OD matrices using the Jensen--Shannon criterion. Based on the conversion of the OD matrix into probabilities $p_{ij}$ defined in Eq. (\ref{['equ_prob']}), the Jensen--Shannon divergence was used to compare OD matrices. (a) Accumulated OD matrices: as additional trajectories are incorporated from OD-2 through OD-6, the matrix progressively captures more information, with OD-3 and OD-4 providing the largest contributions. (b) Jensen--Shannon divergence across the four semesters of the study, showing subtle temporal changes, with smaller differences observed between consecutive semesters.
• Figure 5: Relationship between the transition probability $\omega_{i \to j}^{(\mathrm{OD})}$ and the geographic distance $d_{ij}$ between zones $i$ and $j$. Analysis of SITP users across semesters: (a) 2023-II, (b) 2024-I, (c) 2024-II, and (d) 2025-I. Bidimensional histograms are constructed from $\log_{10}\omega_{i\to j}^{(\mathrm{OD})}$ and $\log_{10}(d_{ij}/d_0)$, with $d_0=1\,\mathrm{m}$ as a reference distance. Frequencies $f(d_{ij},\omega_{i\to j}^{(\mathrm{OD})})$ are encoded in the color bar, showing hexagonal bin counts on a logarithmic scale.