Unravelling the spatial directionality of urban mobility

Human mobility underpins applications from disease control and traffic forecasting to urban planning. Advances in data (e.g., currency flow, GPS, mobile phone records) have enabled high-resolution insights into mobility. Prior work established fundamental laws and models at individual and population scales, often using origin-destination (OD) matrices without explicit spatial orientation. Recent metrics derived from non-spatial OD matrices (e.g., hotspot flow matrices, flow hierarchies) characterize urban mobility structure but do not capture spatial directionality—how flows are oriented across directions relative to urban landmarks such as the city centre. This spatial directionality is critical: for example, a shift of the CBD would be invisible to non-spatial OD analysis yet would substantially alter directional flows. The paper addresses this gap by proposing vector-based metrics that combine magnitude and direction of flows to quantify spatial directionality. Using mobile phone data from ~90 million users across 60 Chinese cities, the study introduces the population mobility vector (PMV) and derives two city-scale metrics—anisotropy and centripetality—to measure directional imbalance and centre-oriented flow, respectively, thereby revealing distinct urban mobility patterns and their implications.

Early research focused on basic mobility characteristics (e.g., fat-tailed jump size distributions) and models for inter-location flows (e.g., radiation model, population-weighted opportunities). More recent studies use non-spatial OD metrics to characterize urban mobility organization, such as hotspot flow matrices and flow hierarchies that reveal stratification across world regions. Traditional urban spatial structure studies rely on population density, infrastructure, and employment distribution. However, these approaches overlook spatial directionality—orientation of flows relative to reference directions (true north) or landmarks (city centre). Vector-field approaches for recurrent mobility and route geometry metrics consider direction and orientation but do not provide city-level summary metrics capturing centripetal tendencies toward the city centre. This work builds on these strands by introducing PMV-based anisotropy and centripetality to quantify spatial directionality and bridge micro-macro perspectives on urban mobility.

Data: Aggregated, anonymised mobile phone records from a Chinese telecommunications operator covering ~90 million users across 60 cities (August and November 2019). Users represent on average 18% of total city populations. The operator partitions cities into 0.005° (~0.5 km × 0.5 km) grids; flows are counts of trips between grid pairs per hour. Records include date, hour, origin and destination coordinates, and trip counts. Only aggregate flows are used.

Preprocessing: Spatially merged to 1 km × 1 km grids. Temporally averaged hourly trips across 25 typical weekdays (Tue–Thu). Morning peak hour 07:00–07:59 selected to quantify overall commuting characteristics and classify cities.

Urban area extraction: Administrative boundaries were harmonized using the Global Human Settlement dataset to delineate urban areas (≥1500 inhabitants/km² or built-up areas with ≥50,000 inhabitants). Analyses focus on flows originating within urban areas.

Metrics and PMV: The OD matrix T_ij records flow from origin i to destination j. Each flow is vectorized as T_ij * u_ij, where u_ij is the unit vector from i to j on a Cartesian plane with y-axis aligned to true north. To mitigate outflow volume heterogeneity, vectors are normalized by total outflow O_i at origin i, yielding unit outflow vectors. The population mobility vector (PMV) at origin i is the vector sum over all destinations: T_i = Σ_j (T_ij / O_i) u_ij. The PMV magnitude and orientation summarize the aggregate direction of outflows from i.

Anisotropy (λ_i, Λ): The PMV magnitude at i, λ_i (0–1), measures directional imbalance of outflows: λ_i = |T_i|. It equals 1 if all outflows share the same direction and 0 if flows are spatially symmetric. City-level anisotropy Λ is the outflow-weighted average: Λ = Σ_i O_i λ_i / Σ_i O_i.

Centripetality (γ_i, Γ): Define a common reference direction using the city centre C. Let u_ic be the unit vector from i to C, and let θ_i be the angle between T_i and u_ic (0 to π). The centripetality at i is γ_i = 1 − θ_i/π (0–1), where higher values indicate stronger orientation toward the city centre. City-level centripetality Γ is the outflow-weighted average: Γ = Σ_i O_i γ_i / Σ_i O_i.

City centre identification (MCA): A maximum centripetality algorithm searches across all locations within the city to find the centre C that maximizes Γ. Steps: (1) choose a candidate centre and compute Γ, (2) repeat for all locations, (3) select the location with maximum Γ as the city centre. For polycentric cities, the primary centre is treated as the most influential for city-wide flows; supplementary analyses identify multiple centres.

Average commuting distance: Proxy by Euclidean distance between grid centroids. City-level average D over the period is D = Σ_ij d_ij t_ij / Σ_ij t_ij.

Mesoscopic ring analysis: Cities are partitioned into l = 5 concentric ring levels around the identified centre with equal trip generation per level, enabling cross-city comparison of spatial profiles of Λ and Γ.

Temporal analysis: Hourly Λ and Γ computed over a day to assess diurnal dynamics across city types.

Microscopic model (RWRC): A random workplace and residence choice model on a 2D l × l lattice (l = 50). Start with the first individual choosing the centre as both workplace and residence. For each added individual (N = 10^4 total): workplace choice at location j occurs with probability proportional to N_j^α, where N_j is active population (residents + workers) and α ≥ 0 captures employment attraction strength (preferential attachment). Residence choice at location i ≠ j follows Q_ij ∝ exp(−d_ij/β), where β > 0 is the commuting distance scale. Simulations vary α in [0, 2] by 0.1 increments and increase β from 0.01 by 0.1 until Λ and Γ stabilize; results averaged over 100 runs. The model outputs PMVs and derived Λ, Γ to compare with empirical city types.

• The PMV-based metrics effectively quantify spatial directionality. Example city metrics at 07:00: Beijing Λ = 0.40, Γ = 0.92 (strong monocentric); Tianjin Λ = 0.28, Γ = 0.87 (weak monocentric-like); Foshan Λ = 0.30, Γ = 0.65 (polycentric).
• Clustering: In anisotropy–centripetality space, hierarchical clustering of 60 cities yields three types: strong monocentric (n = 36), weak monocentric (n = 13), polycentric (n = 11). Differences are statistically significant: • Anisotropy comparisons (two-sided t-tests, Bonferroni-corrected): Type1 vs Type2 P = 9.43 × 10^−6; Type2 vs Type3 P = 0.003; Type1 vs Type3 P = 7.96 × 10^−7. • Centripetality comparisons: Type1 vs Type2 P = 2.40 × 10^−6; Type2 vs Type3 P = 4.70 × 10^−4; Type1 vs Type3 P = 2.60 × 10^−7.
• Commuting distance vs city size: • Against urban area A: D ~ log(A)^a. Strong monocentric: a = 3.67, R^2 = 0.87, P = 8.91 × 10^−17; weak monocentric: a = 2.86, R^2 = 0.80, P = 3.52 × 10^−5; polycentric: a = 0.87, R^2 = 0.17, P = 0.21 (not significant). Thus, commuting distance grows with area in monocentric cities but not in polycentric cities. • Against urban population N: D ~ log(N)^b. Strong monocentric: b = 2.70, R^2 = 0.80, P = 2.13 × 10^−13; weak monocentric: b = 2.04, R^2 = 0.82, P = 2.10 × 10^−5; polycentric: b = 1.07, R^2 = 0.22, P = 0.15 (not significant).
• Mesoscopic spatial profiles: • Anisotropy increases with distance from the centre for all types. Increase from core to periphery: strong monocentric +80.3%, weak monocentric +75.5%, polycentric +40.5%. Polycentric cities show relatively stable anisotropy beyond outer levels (~Level 12), indicating balanced job-direction distribution in peri-urban areas. • Centripetality declines with distance for all types: strongest decline in polycentric (−22.1%), then weak monocentric (−13.2%), lowest in strong monocentric (−2.1%). Monocentric cities exhibit lower centripetality within the CBD ring (Level 11) due to abundant local job choices reducing strict centre orientation.
• Temporal dynamics (hourly): Off-peak (09–17 h) Λ and Γ are relatively stable across all types. Evening (18–21 h) both metrics decrease (more isotropic and centrifugal homeward flows). Late night (post-22 h) anisotropy rises while centripetality falls (anisotropic, centrifugal leisure flows). Morning peak (04–08 h): centripetality increases most in strong monocentric (+23.6%), then weak monocentric (+22.0%), least in polycentric (+15.4%). Anisotropy rises during the peak in strong monocentric, remains stable in weak monocentric, and decreases in polycentric. Polycentric cities often show higher anisotropy than weak monocentric throughout the day due to consistent directional pulls of multiple subcentres.
• Microscopic mechanism: The RWRC model reproduces the three empirical patterns by tuning employment attraction strength α and commuting distance scale β. High α and β yield strong monocentric patterns (high Λ, Γ); moderate values yield weak monocentric (low Λ, high Γ); low values yield polycentric (low Λ, low Γ). This links individual workplace/residence choices to macroscopic directional mobility patterns.

The study formulates a vector-based framework that captures spatial directionality in urban mobility, overcoming limitations of non-spatial OD analyses and hotspot-based methods. By summarizing flows into PMVs and city-level anisotropy and centripetality, the approach reveals how directional imbalances and centre-oriented tendencies structure commuting. These metrics successfully differentiate city types and explain how scaling of average commuting distance depends on urban form: monocentric cities experience longer commutes as they grow in area and population, while polycentric cities maintain relatively stable commutes due to mixed and distributed employment-residence patterns. Mesoscopic and temporal analyses further elucidate core-periphery dynamics and diurnal rhythms of directional flows, offering actionable insights for transport planning (e.g., reinforcing radial capacity in monocentric cores, balancing peripheral job locations). The RWRC model provides a plausible behavioral mechanism, showing how the interplay of employment attraction and distance sensitivity can generate observed directional patterns. Overall, the findings highlight spatial directionality as an essential dimension for understanding and managing urban mobility, with implications for congestion mitigation, suburban development, and sustainable urban growth.

This work introduces PMV-based anisotropy and centripetality as compact, interpretable metrics to quantify spatial directionality of urban mobility from large-scale mobile phone data. Applying the method to 60 Chinese cities identifies three robust mobility archetypes (strong monocentric, weak monocentric, polycentric) with distinct scaling of commuting distances, spatial profiles of directional imbalance and centre orientation, and diurnal dynamics. A simple RWRC model links micro-level workplace and residence choices to macro-level directional patterns, highlighting employment attraction strength and commuting distance scale as key drivers. These contributions provide a practical diagnostic toolkit for urban planners and transport policymakers to assess and guide mobility patterns. Future work could: (1) extend centre identification to multi-centre detection and dynamic centres; (2) incorporate congestion, housing markets, and socio-economic heterogeneity into behavioral models; (3) validate across diverse countries and data sources; and (4) refine distance measures (e.g., network-based travel times) to improve realism and policy relevance.

• The RWRC model is stylized and omits important determinants such as traffic congestion, housing prices, and socio-economic factors, which may influence workplace and residence choices and thus directional flows.
• City centre identification via the maximum centripetality algorithm selects a single optimal centre for analysis; although supplementary work identifies multiple centres, automated robust multi-centre detection warrants development.
• Commuting distance is approximated by Euclidean distance between grid centroids, which may differ from network travel distances and times.
• The mobile phone dataset is aggregated and anonymised; data access is restricted due to privacy and contractual limitations, and users represent about 18% of city populations, potentially introducing sampling bias.
• Ring-based mesoscopic analysis assumes equal trip generation per level and may abstract from fine-grained urban morphology differences across cities.