Universal scaling laws of collective human flow patterns in urban regions

The study of human movement patterns has a long history, initially relying on population data and surveys. Early research established empirical laws of migration (Ravenstein, 1885), but data limitations restricted analysis to macroscopic scales or focused on car flows (Helbing, 2001). The advent of mobile phones with GPS has revolutionized this field (Ratti et al., 2006; Karamshuk et al., 2011; Blondel et al., 2015; Wang et al., 2018), providing detailed, real-time location data for vast populations. Research now broadly divides into studies of individual trajectories (Zhao et al., 2015; Jurdak et al., 2015; Alessandretti et al., 2017) and global migration between cities (Ren et al., 2014; Yan et al., 2014). Individual trajectories, while seemingly random, differ from Brownian motion, often exhibiting Levy flight characteristics (González et al., 2008). Studies also classify trajectories by social activity (Schneider et al., 2013; Pappalardo et al., 2015; Jiang et al., 2017) and examine trajectory predictability (Song et al., 2010; Cuttone et al., 2018). Specific events like earthquakes (Lu et al., 2012; Hara & Kuwahara, 2015) and traffic congestion resilience (Zhang et al., 2019) have also been analyzed. Macroscopically, the gravity law and intervening opportunity models have been applied to human migration (Jung et al., 2008; Thiem Thiemann et al., 2010; Barthélemy, 2014; Simini et al., 2012; Noulas et al., 2012; Yan et al., 2017; Liu & Yan, 2019; Liu & Yan, 2020), alongside probabilistic prediction for congestion and advertising (Krumm & Horvitz, 2006; Kim et al., 2011; De Brébisson et al., 2015; Besse et al., 2017). City potential has been estimated using vector fields from origin-destination matrices (Mazzoli et al., 2019), but mesoscopic analysis within cities remains under-explored. This study aims to address this gap by analyzing collective human flow within urban areas using a novel framework.

Existing research on human mobility uses various approaches. Microscopic studies analyze individual movement patterns, often revealing non-random characteristics such as Levy flights. Macroscopic studies focus on migration between cities, employing models like the gravity model and intervening opportunity models. However, mesoscopic analysis, focusing on collective flow patterns within cities, is less prevalent. This paper contributes to this area by introducing a new framework for analyzing collective human mobility at the mesoscopic scale within urban areas.

The research utilized GPS location data from approximately 260,000 mobile phone users in Japan, provided by Agoop. Data included user ID, date, time, longitude, latitude, and velocity components. Data collection occurred daily (excluding 1 a.m. to 5 a.m.) at approximately 30-minute intervals. To analyze collective flow, the researchers employed a coarse-graining method: 1. **Velocity Discretization:** The urban map was divided into 500m x 500m squares. The average velocity vector in each square was calculated for 30-minute intervals, assigning one of four directional vectors (North, East, South, West) based on the dominant velocity component. 2. **Drainage Basin Identification:** A drainage basin was defined as a connected set of squares, where the flow direction in each square pointed towards a neighboring square within the same basin. This method enabled unique identification of drainage basins. 3. **Data Analysis:** The researchers analyzed drainage basin size distributions, the number of moving people in each basin, and the relationship between basin size and the number of moving people. They employed the Kolmogorov-Smirnov test to assess power-law distributions, using maximum likelihood estimation to determine power-law exponents. Additionally, they analyzed office floor area and daytime worker population data for Tokyo to explore further scaling relations. The distance from the city center (Imperial Palace) served as a key variable in these analyses. Specifically, weighted averages were computed to account for varying GPS transmission frequencies among users. Equations (14) to (17) describe the calculation of weighted probabilities and average velocities within each square. The Kolmogorov-Smirnov (KS) test was employed to validate the power law distribution hypothesis. The procedure involved defining null and alternative hypotheses, estimating power law exponents using maximum likelihood estimation, determining the optimal xmin, calculating the KS statistic D, generating random datasets, and computing the p-value. Tables 1 and 2 present the results of the KS test for basin size and population in basins across the nine cities.

The study revealed several significant scaling laws governing collective human flow in urban areas: 1. **Morning Rush Hour Dominance:** During morning rush hour, strong flows towards the city center created large drainage basins, while afternoon patterns were more random and exhibited smaller basin sizes. This was consistent across all nine cities. 2. **Power Law Distributions:** The cumulative distribution functions (CDFs) of basin sizes during the morning rush hour followed a power law with an exponent of approximately -2.4, whereas afternoon distributions were approximately exponential, reflecting the random nature of movement at that time. This power-law behavior was consistent across the nine cities. 3. **Three-Dimensional Scaling:** The most striking finding was the three-dimensional scaling relation between the number of moving people (p) in a basin and its diameter (Lb): p ∝ Lb³. This contradicts the intuitive expectation of a two-dimensional relationship (p ∝ Lb²) and suggests a concentration of movement in a three-dimensional structure, likely influenced by skyscrapers and public transport infrastructure focused on the city centre. Relatedly, the population of moving people was shown to be proportional to the square of the basin size (p ∝ Sb²). 4. **Fractal Geometry of Basins:** The researchers found a fractal dimension of approximately 1.5 for the basin structure. The population density within basins followed a power law decay with distance from the center (r⁻⁰·⁵). 5. **Tokyo Case Study:** Analysis of office floor area and daytime worker population in Tokyo corroborated the findings. Office floor area and daytime population density showed inverse power-law relationships with distance from the city center, suggesting a high concentration of activity in the central areas.

The findings challenge conventional understandings of human flow in urban regions. The unexpected three-dimensional scaling relation during morning rush hour, characterized by p ∝ L³, highlights the influence of city structure and high-capacity infrastructure like skyscrapers and centralized public transportation. The power-law distributions of basin sizes and the fractal geometry of drainage basins suggest self-organized criticality, a common feature of complex systems. The study's findings are consistent across nine major Japanese cities, suggesting the existence of universal scaling laws governing collective human flow in urban environments. These patterns likely hold for cities globally, although further research across diverse urban contexts would strengthen these conclusions.

This study provides new insights into the fundamental scaling laws governing collective human flow in urban environments. The discovery of the three-dimensional scaling relation (p ∝ L³), the power-law distributions of basin sizes, and the fractal geometry of drainage basins during peak commuting hours provides a robust foundation for understanding city dynamics. Future research could extend this framework to analyze other cities worldwide and investigate the impact of diverse urban planning strategies and socioeconomic factors on these universal patterns. Furthermore, exploring the effects of extreme events, such as natural disasters, and the incorporation of additional data sources (e.g., social media) would yield valuable insights.

The study is primarily based on data from Japan. While the findings suggest the existence of universal scaling laws, further research across diverse geographical locations and cultural contexts is needed to confirm generalizability. The coarse-graining method employed for velocity discretization may introduce some level of simplification, potentially influencing the precision of the results. The data privacy concerns imposed limitations on the data's accessibility.