Defining the urban "local" with low dimensional manifolds of human mobility networks

TL;DR

The paper tackles the challenge that universal urban models obscure local heterogeneity by defining locality in human mobility networks through a topological lens. It proves that mobility localities map to geographic localities and that the resulting mobility networks lie on low-dimensional manifolds with dimension $d\le 5$, enabling compact spatial embeddings. The authors develop a practical embedding pipeline using a hyperbolic distance, Topologically Constrained Isometric Embedding (TCIE), and demonstrate two key applications: location choice and propagation modelling, showing uniform facility layouts and isotropic diffusion patterns on the mobility manifolds. Across five diverse sites, the framework bridges geography and network science, offering a scalable, geometry-based toolkit for urban analysis with clear operational benefits for planning and public health.

Abstract

Urban science has largely relied on universal models, rendering the heterogeneous and locally specific nature of cities effectively invisible. Here we introduce a topological framework that defines and detects localities in human mobility networks. We empirically demonstrate that these human mobility network localities are rigorous geometric entities that map directly to geographic localities, revealing that human mobility networks lie on manifolds of dimension <=5. This representation provides a compact theoretical foundation for spatial embedding and enables efficient applications to facility location and propagation modeling. Our approach reconciles local heterogeneity with universal representation, offering a new pathway toward a more comprehensive urban science.

Figures (8)

• Figure 1: The locality of human mobility.a, Localities of human mobility on a map of an undisclosed Japanese city. Irregular regions represent localities derived from normalized mobility flow data. Highlighted colored localities are examples from the pale blue regions, shown to trace correspondences between a and b and illustrate intersections. Each locality includes a central node, other nodes, and local edges connecting them. Edge thickness in both a and b is linearly normalized by manifold distance. b, Localities on the human mobility manifolds. Colors correspond to geographic counterparts in a, demonstrating that network localities map to geographic localities. To highlight correspondences, the road network and waterways are linearly mapped using right-angled triangles formed by the nodes, giving the appearance of a distorted map. c, Intersection network of localities. Nodes correspond to the same colored localities in a and b, now condensed to dots, and edges represent locality intersections. This network mirrors the intersection structures in both geographic and manifold spaces, showing that continuous percolation in 2D geographic space is equivalent to percolation on the mobility network, underpinning the manifold’s existence.
• Figure 2: Empirical evidence for the two sufficient conditions guaranteeing the existence of human mobility manifolds. The plots in the left column illustrate the dichotomy of $B$, serving as empirical evidence for the existence of locality. Points along the diagonal represent $B^1$, while those clustered along the upper horizontal line represent $B^2$. The lengths of ${OB}^2$ are all 100 because, for computational convenience, the distances between unconnected node pairs are artificially set to a sufficiently large number, which is 100 in our practice, to approximate what would theoretically be infinite distances. The plots in the right column provide empirical evidence for the second condition: when measuring the geographic distances between the two kinds of $B$ and $O$, we find ${{OB}^1}_g<{{OB}^2}_g$, namely those $B$ outside the network locality at $O$ are also geographically far away to $O$.
• Figure 3: The embedding quality of human mobility manifolds, measured both topologically and metrically. The plots in the left column represent the topological embedding quality. We select node pairs whose embedding distances fall within an increasingly large distance window (with window size as $x$-axis) and examine two metrics: the proportion of these pairs that are actual local edges in the original graph (orange curve), and the proportion of all local edges that are covered by these selected node pairs (blue curve). A higher intersection point of the two curves indicates better topological embedding quality. The heights of the intersections are 0.78, 0.69, 0.92, 0.81, and 0.90 for subfigures a–e, respectively. The plots in the right column represent the metric embedding quality. The Pearson correlation coefficients for metric embedding in subfigures a–e are, 0.87, 0.75, 0.92, 0.98, and 0.81, respectively.
• Figure 4: The location choice problem and the propagation problem on the human mobility manifolds.a-c, The location choice problem (the case of Shenzhen). The node colors in b, c depict $\log_{10} \|c_i\|$, where $c_i$ is the set of individuals covered by the grid (namely node), see Methods for the definition of coverage; the cross marks depict optimal location solutions. The classic location choice law under the featureless plain assumptionChristaller1933 gives rise to uniformly laid location patterns a, which are achieved with real-world data in b. The uniformity of optimal location solutions on the human mobility manifolds is evident both quantitatively (see Methods) and when compared with that on the geographic space c. d-f, the propagation problem (the case of Beijing). The node size in e, f depicts the number of infected cases in that cell. The classic propagation law would predict a concentric diffusion pattern in the featureless space d, which once again, is achieved with real-world data in e. The concentric test is described in the Methods, and is also evident when compared with the propagation on the geographic space f.
• Figure ED5: Human mobility manifolds in England at finer spatial granularity. Empirical and embedded manifolds of human mobility in England, constructed with one finer level of H3 hexagonal grids (4,840 nodes in total) than in the main analysis. a, Dichotomy analysis for the existence of localities. b, Empirical evidence on mapping between network and geographic localities. c, Topological quality of the embedding, with intersection point at 0.8684. d, Metric quality of the embedding, with Pearson correlation coefficient $r = 0.8259$.