Identifying and Characterising Higher Order Interactions in Mobility Networks Using Hypergraphs

TL;DR

This work addresses the limitation of pairwise mobility representations by introducing co-visitation hypergraphs to capture higher-order, group-level location interactions within temporal observation windows. It constructs the hypergraph by segmenting time into overlapping windows of length $\Delta T$, forming transactions per individual, mining frequent location itemsets with FP-Growth under a threshold $min\_sup$, and treating each frequent itemset as a weighted hyperedge with weight $\mathbf{S}(e)$. Applying the framework to city-scale mobility data, the authors analyze structural features (e.g., exponential degree distributions, increasing hyperedge sizes with $\Delta T$) and spatial patterns, compare regular and emergency days, and link POI density to node degree through a power-law relationship. The study demonstrates the utility of hypergraph-based mobility analysis for urban planning, public health, and disaster response, while acknowledging data-granularity and scope limitations and providing open-source code for reproducibility.

Abstract

Understanding human mobility is essential for applications ranging from urban planning to public health. Traditional mobility models such as flow networks and colocation matrices capture only pairwise interactions between discrete locations, overlooking higher-order relationships among locations (i.e., mobility flow among two or more locations). To address this, we propose co-visitation hypergraphs, a model that leverages temporal observation windows to extract group interactions between locations from individual mobility trajectory data. Using frequent pattern mining, our approach constructs hypergraphs that capture dynamic mobility behaviors across different spatial and temporal scales. We validate our method on a publicly available mobility dataset and demonstrate its effectiveness in analyzing city-scale mobility patterns, detecting shifts during external disruptions such as extreme weather events, and examining how a location's connectivity (degree) relates to the number of points of interest (POIs) within it. Our results demonstrate that our hypergraph-based mobility analysis framework is a valuable tool with potential applications in diverse fields such as public health, disaster resilience, and urban planning.

Figures (7)

• Figure 1: Hypergraph Example.A) The hypergraph $H = (V, E)$ consists of 6 vertices and 5 hyperedges. The vertex set is $V = \{u_1, u_2, u_3, u_4, u_5, u_6\}$ and the hyperedge set is $E = \{\{e_1 = \{u_1, u_2\}, e_2 = \{u_1, u_4, u_5\}, e_3 = \{u_4, u_5\}, e_4 = \{u_2, u_3\}, e_5 = \{u_2, u_5, u_6\}\}$. The vertices are represented as circles, and the hyperedges are depicted as colored regions encompassing the vertices involved in each hyperedge. B) The corresponding bipartite graph of $H$ is shown, where $V$ forms one partition and $E$ forms the other. The edges in the bipartite graph are colored according to the hyperedge colors from the original hypergraph. C) The incidence matrix $I = \{I_{ij}\}$ is shown, where $I_{ij} = 1$ if hyperedge $e_j$ is incident on vertex $u_i$. The matrix is accompanied by the vertex degrees, calculated as the sum of the entries in each row. The vertices $u_2$ and $u_5$ have the highest degree (3), while $u_3$ and $u_6$ have the lowest degree (1). D) The 3-uniform subhypergraph of $H$ includes two hyperedges of cardinality 3: $e_2 = \{u_1, u_4, u_5\}$ and $e_5 = \{u_2, u_5, u_6\}$, with their intersection being the vertex $u_5$.
• Figure 2: Schematic of constructing a co-visitation hypergraph from individual-level human movement data.A) (left) Sample trajectories for three individuals, $u_1$, $u_2$, and $u_3$, over a single day, shown on a 3x3 grid map in blue, red, and green. (right) These trajectories are summarized as sets of visited locations: $\{\ell_1, \ell_2, \ell_4, \ell_5\}$ for $u_1$, $\{\ell_4, \ell_5, \ell_6, \ell_7\}$ for $u_2$, and $\{\ell_4, \ell_8\}$ for $u_3$. B) A sliding time window of length $\Delta T = 3$ days moves across a sequence of $D$ days, creating observation windows ${w_1, w_2, \dots, w_{D-2}}$. For representative windows $w_1$, $w_2$ shown, and $w_{D-2}$, each circle on the grid represents a visit by one of the three individuals to a location, with colors indicating which individual visited (blue for $u_1$, red for $u_2$, and green for $u_3$). C) In each observation window, the locations visited by an individual are treated as a transaction, with the locations forming the set of items. This converts mobility data into a transaction dataset $\mathcal{T} = \{T_1, \cdots, T_M\}$, allowing the identification of frequently co-visited locations across $M$ individuals over $D$ days. Potential candidates for frequently co-visited locations (frequent itemsets) are highlighted: $\{\ell_1, \ell_2\}$ and $\{\ell_4, \ell_5, \ell_6\}$. D) The FPGrowth algorithm takes the transaction dataset $\mathcal{T}$, a minimum support threshold $min\_sup$ (set to 0.05 in this example, meaning an itemset must appear in at least 5% of transactions to be considered frequent), and a minimum itemset size $min\_size$ (set to 2) as inputs. It identifies all frequent itemsets that meet these criteria. The table on the right displays the results for the example, showing 9 frequent itemsets of size 2 and 2 frequent itemsets of size 3. E) The mobility hypergraph is constructed from the frequent itemsets identified in the previous step, where locations serve as nodes and frequent itemsets form the hyperedges. In this example, location $\ell_5$ has the highest degree (7), while locations $\ell_1$, $\ell_3$, and $\ell_9$ have the lowest degree (1). Both hyperedges of size 3 are incident on $\ell_5$.
• Figure 3: Structural and Spatial Properties of the Co-Visitation Hypergraph.A) CCDF of node degrees for hypergraphs with $min\_sup = 0.005$ and $\Delta T \in \{1, 3, 7\}$ (green, blue, and red, respectively). The x-axis shows node degree, and the y-axis shows $P(\text{Degree} \geq k)$. B) Hyperedge size distribution for $\Delta T \in \{1, 3, 7\}$, with the x-axis representing hyperedge size and the y-axis representing frequency in log-scale. The maximum hyperedge size is 4 for $\Delta T = 1$, and 5 and 7 for $\Delta T = 3$ and $\Delta T = 7$. C) Maximum chebyshev distance over $\Delta T$ values. X-axis represents $\Delta T$ and y-axis represents the max chebyshev distance. Each curve corresponds to a particular $min\_sup$ value. D,E,F) Spatial heatmaps of node degrees for hypergraphs with $min\_sup = 0.005$ and $\Delta T \in \{1, 3, 7\}$ on a 20x20 grid. Nodes are represented as grid cells according to their $(x, y)$ coordinates, with darker shades of red indicating higher node degrees.
• Figure 4: Comparison of hypergraphs capturing mobility patterns on regular and emergency days.A) Hyperedge size distribution for $\Delta T$ values at $min\_sup = 0.005$. The x-axis shows hyperedge size, and the y-axis (log scale) represents frequency. Blue curves denote regular days, red curves emergency days, with markers for $\Delta T$. Larger hyperedges are more frequent on regular days, with differences increasing at higher $\Delta T$. B) Maximum Chebyshev distance for higher-order interactions (hyperedges $\geq 3$) across $\Delta T$ and $min\_sup$. Solid bars show regular days, striped bars emergency days, with colors for $min\_sup$ values. Differences peak at $min\_sup = 0.005$, suggesting reduced long-distance movement during emergencies. C--D) Unique edges in co-degree graphs for regular and emergency hypergraphs at $\Delta T = 3$, $min\_sup = 0.005$ and $\Delta T = 7$, $min\_sup = 0.005$. Blue edges are unique to regular days, while red edges are unique to emergency days. Edge thickness reflects co-occurrence frequency. These visualizations highlight shifts in co-visitation patterns during emergencies.
• Figure 5: POI Distribution Among Locations and Its Relationship with Degree Centrality in Hypergraphs. A) Heatmap showing the total number of Points of Interest (POI) at each location, represented by X and Y coordinates. Darker shades of red indicate higher POI counts. B) Clustering of locations obtained using K-means clustering. Each location is represented by a vector corresponding to the number of POIs of each category. The similarity between these POI vectors is used as the metric, and colors represent cluster membership. C) Scatter plots depicting the correlation between the total POI count at each location and the corresponding degree centrality in hypergraphs generated across varying temporal window lengths ($\Delta T$). The x-axis represents the degree of a location in hypergraph for given $\Delta T$ and $min\_sup=0.015$, while the y-axis represents the total POI count in that location. The dashed black line represents the best power-law fit to the data.