Identifying and Characterising Higher Order Interactions in Mobility
Networks Using Hypergraphs
PRATHYUSH SAMBATURU, Department of Biology & Pandemic Sciences Institute, University of Oxford,
United Kingdom
BERNARDO GUTIERREZ, Department of Biology & Pandemic Sciences Institute, University of Oxford,
United Kingdom
MORITZ U.G. KRAEMER, Department of Biology & Pandemic Sciences Institute, University of Oxford,
United Kingdom
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.
CCS Concepts: •Mathematics of computing →Hypergraphs ;•Information systems →Data mining ;•Applied
computing ;
Additional Key Words and Phrases: hypergraphs, higher-order interactions, frequent pattern mining, human mobility data
1 Introduction
Human mobility data is crucial for understanding patterns of movement across geographical regions, with
applications spanning urban planning[ 1], transportation systems design[ 2], infectious disease modeling and
control [ 3,4], and social dynamics studies [ 5]. Traditionally, mobility data has been represented using flow
networks[ 6,7] or colocation matrices [ 8], where the primary representation is via pairwise interactions. In flow
networks, this means directed edges represent the movement of individuals between two locations; colocation
matrices measure the probability that a random individual from a region is colocated with a random individual
from another region at the same location. These data types and their pairwise representation structure have been
used to identify the spatial scales and regularity of human mobility, but have inherent limitations in their capacity
to capture more complex patterns of human movement involving higher-order interactions between locations –
that is, group of locations that are frequently visited by many individuals within a period of time (e.g., a week)
and revisited regularly over time. Higher-order interactions between locations can contain crucial information
under certain scenarios. For example, in contact tracing during the early stages of an epidemic, identifying these
Authors’ Contact Information: Prathyush Sambaturu, Department of Biology & Pandemic Sciences Institute, University of Oxford, Oxford,
United Kingdom, prathyush.sambaturu@biology.ox.ac.uk; Bernardo Gutierrez, Department of Biology & Pandemic Sciences Institute,
University of Oxford, Oxford, United Kingdom, bernardo.gutierrez@biology.ox.ac.uk; Moritz U.G. Kraemer, Department of Biology &
Pandemic Sciences Institute, University of Oxford, Oxford, United Kingdom, moritz.kraemer@biology.ox.ac.uk.
, Vol. 1, No. 1, Article . Publication date: March 2025.arXiv:2503.18572v1  [cs.SI]  24 Mar 2025
2 • Prathyush Sambaturu, Bernardo Gutierrez, and Moritz U.G. Kraemer
higher-order interactions of locations over a period of, say, three days enables health authorities to trace all
locations visited by potentially exposed individuals during that time. Pairwise representations of mobility data
are not useful for this type of analysis. Addressing these limitations requires a more expressive representation of
mobility data.
Unlike simple graphs, which model pairwise interactions, hypergraphs [ 9] can represent higher-order rela-
tionships, where a single edge (hyperedge) connects an arbitrary number of nodes. Hypergraphs have been
employed in various domains, such as epidemic modeling[ 10,11] and biological systems[ 12], where higher-order
interactions between entities play a significant role in the spread, maintenance, and stability of these systems.
In this paper we propose a novel approach to construct co-visitation hypergraphs from individual-level mobility
trajectory data such as those available from eXtended Data Records (XDR) or GPS traces[ 13]. Our method
captures higher-order interactions of locations while allowing the study of mobility dynamics across varying
temporal and spatial scales through a parameter called the observation window length (Δ𝑇). Temporal windows
divide the timeline into overlapping segments, allowing insights into how group interactions evolve over daily
or weekly periods. We also develop analyses to understand the spatio-temporal mobility behaviors from the
constructed hypergraphs. Our framework facilitates a detailed examination of the structural and spatial properties
of hypergraphs and their change across temporal windows and during periods characterised by business-as-usual
behaviour as well as external shocks.
Recent studies relevant to hypergraph-based mobility analysis focus mainly on hypergraph embedding ap-
proaches [ 14][15]. Yang et al. 2019 [ 14] embed user mobility and social relationships using hypergraphs, linking
users, locations, times, and activities to perform tasks such as friendship and location prediction. In contrast,
our work focuses on locations rather than users, constructing co-visitation hypergraphs to capture higher-order
interactions between locations visited together within specific temporal windows ( Δ𝑇). This location-centric per-
spective is particularly useful for analyzing changing mobility behaviors across regions and during emergencies,
such as natural disasters or infectious disease outbreaks.
We summarize our contributions in this work as follows:
•Temporal window-based co-visitation hypergraph : We introduce a novel hypergraph-based representa-
tion of mobility data that captures complex patterns of human movement from a location-centric perspective.
•Algorithm to construct co-visitation hypergraph : We introduce a flexible algorithm to construct co-
visitation hypergraphs from individual-level mobility data using temporal observation windows ( Δ𝑇), which
can enable analysis of co-visitation patterns over varying time scales.
•Application to individual-Level mobility data : We validate our approach using YJMob100K[ 16] dataset
performing the following analysis.
–Structural and spatial analysis . We analyze the structural and spatial properties of co-visitation hypergraphs
to characterize each location’s role in group interactions, determining whether it is central or peripheral
and whether it is co-visited with nearby or distant locations. Further, we investigate how our algorithmic pa-
rameters influence key hypergraph characteristics. Specifically, we investigate hyperedge size distributions,
spatial coverage, and the evolution of higher-order interactions.
–Framework to examine mobility changes during external shocks . We extend the analysis to study changes
in mobility patterns during emergency scenarios using a pre-defined phase in the YJMob100K dataset. By
comparing “regular” and “emergency” days, we identify significant differences in hyperedge structures,
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Identifying and Characterising Higher Order Interactions in Mobility Networks Using Hypergraphs • 3
spatial coverage, and interaction patterns. This framework demonstrates the utility of co-visitation hyper-
graphs for analyzing disruptions in mobility, such as those caused by extreme weather events or public
health crises, offering insights into shifts in co-visitation dynamics under such conditions.
Our framework has potential applications in studying mobility behaviors during emergencies, disasters, and
provide the potential interaction network for modelling infectious diseases. Further, outputs could be used to
understand the evolution of co-visitaiton hypergraphs, how different locations emerge as central nodes and those
that lose their centrality.
2 Preliminaries
In this section, we provide the necessary background to clarify the rest of the paper.
2.1 A Brief Overview of Hypergraphs
A hypergraph is denoted by 𝐻=(𝑉,𝐸), where𝑉={𝑢1,···,𝑢𝑛}is a set of vertices (or nodes), and 𝐸is a family
of subsets of 𝑉called hyperedges. A hyperedge 𝑒𝑗={𝑢𝑗1,...,𝑢𝑗𝑙}∈𝐸generalizes the concept of an edge in a
simple graph, where |𝑒𝑗|=𝑙for𝑙≥0. In the special case of a simple graph, each edge has exactly two vertices
(𝑙=2). Hypergraphs, however, allow for higher-order interactions among entities, unlike simple graphs which
are limited to pairwise interactions. Figure 1A shows an example hypergraph with six vertices (or nodes) and
five hyperedges.
A hypergraph 𝐻=(𝑉,𝐸)can be represented as a bipartite graph 𝐵=(𝑉∪𝐸, 𝐸𝐵), with two partitions: the set of
vertices𝑉in one partition and the set of hyperedges 𝐸, which are represented as nodes in the second partition.
In the bipartite graph 𝐵, an edge(𝑢𝑖,𝑒𝑗)∈𝐸𝐵exists if and only if the hyperedge 𝑒𝑗in𝐻is incident on the vertex
𝑢𝑖. The bipartite representation of the hypergraph from Figure 1A is illustrated in Figure 1B.
The incidence matrix of a hypergraph 𝐻is denoted by 𝐼𝑛×𝑚={𝐼𝑖𝑗}(𝑚denotes number of hyperedges in 𝐻),
where𝐼𝑖𝑗=1if hyperedge 𝑒𝑗is incident on node 𝑢𝑖in𝐻, and𝐼𝑖𝑗=0otherwise. The degree of a vertex,𝑢𝑖∈𝑉, is
the number of hyperedges incident on that vertex, calculated as deg(𝑢𝑖)=Í
𝑗𝐼𝑖𝑗. Figure1C shows the incidence
matrix and the degree of each node vertex for the example hypergraph.
The size of a hyperedge is defined as the number of vertices it contains. For example, the size of hyperedge
𝑒2={𝑢1,𝑢4,𝑢5}in the hypergraph 𝐻shown in Figure 1 is 3. The rank 𝑟(𝐻)of a hypergraph is the maximum size
of any hyperedge in 𝐻. In the example shown in Figure 1, the hypergraph 𝐻has three hyperedges ( 𝑒1,𝑒3, and𝑒4)
of size 2, and two hyperedges ( 𝑒2and𝑒5) of size 3. Therefore, the rank of the hypergraph 𝐻,𝑟(𝐻), is 3, since the
largest hyperedges consists of three vertices.
A𝑘-uniform hypergraph is a hypergraph in which all hyperedges have size 𝑘. A𝑘-uniform subhypergraph of a
hypergraph 𝐻is a subhypergraph induced by all hyperedges in 𝐻of size exactly 𝑘. Specifically, for a 3-uniform
subhypergraph 𝐻′=(𝑉′,𝐸′), we have𝐸′⊆𝐸, where𝐸′={𝑒∈𝐸:|𝑒|=3}. The 3-uniform subhypergraph of our
example hypergraph (shown in Figure1D) contains only two hyperedges 𝑒2={𝑢1,𝑢4,𝑢5}and𝑒5={𝑢2,𝑢5,𝑢6}
which intersect at vertex 𝑢5.
2.2 Existing Representations of Human Mobility Based on Location-to-Location Interactions
Empirical human mobility patterns can be inferred from various sources, such as mobile phones, which is
commonly available in the form of Call Detail Records (CDR), eXtended Detail Records (XDR), or GPS traces [ 13].
XDR and GPS trace data typically offer detailed mobility trajectories for individuals within a specific geographical
region along with corresponding timestamps. This means that not only pairwise exchanges between locations
can be inferred by the sequential visitation of locations, including the approximate time of stay in each.
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4 • Prathyush Sambaturu, Bernardo Gutierrez, and Moritz U.G. Kraemer
𝑢!𝑢"𝑢#𝑢$𝑢%𝑢&𝒆𝟏𝒆𝟐𝒆𝟑𝒆𝟒𝒆𝟓BA
Ce1e2e3e4e5deg(v)u1110002u2100113u3000101u4011002u5011013u6000011𝑢!𝑢"𝑢#𝑢$𝑢&𝑢%𝑒!𝑒"
𝑒%𝑒#𝑒$DIncidence Matrix 𝐼'×) and node degrees𝐻𝑦𝑝𝑒𝑟𝑔𝑟𝑎𝑝ℎ	𝐻	=	(𝑉,𝐸)	𝐵𝑖𝑝𝑎𝑟𝑡𝑖𝑡𝑒	𝑅𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛	𝑜𝑓	𝐻
3-uniform subhypergraph of 𝐻𝑢!𝑢"𝑢#𝑢$𝑢%𝒆𝟐𝒆𝟓
Fig. 1. Hypergraph Example. A) The hypergraph 𝐻=(𝑉,𝐸)consists of 6 vertices and 5 hyperedges. The vertex
set is𝑉={𝑢1,𝑢2,𝑢3,𝑢4,𝑢5,𝑢6}and the hyperedge set is 𝐸={{𝑒1={𝑢1,𝑢2},𝑒2={𝑢1,𝑢4,𝑢5},𝑒3={𝑢4,𝑢5},𝑒4=
{𝑢2,𝑢3},𝑒5={𝑢2,𝑢5,𝑢6}}. 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 𝐻is shown,
where𝑉forms one partition and 𝐸forms the other. The edges in the bipartite graph are colored according to the
hyperedge colors from the original hypergraph. C) The incidence matrix 𝐼={𝐼𝑖𝑗}is shown, where 𝐼𝑖𝑗=1if hyperedge
𝑒𝑗is incident on vertex 𝑢𝑖. The matrix is accompanied by the vertex degrees, calculated as the sum of the entries in
each row. The vertices 𝑢2and𝑢5have the highest degree (3), while 𝑢3and𝑢6have the lowest degree (1). D) The
3-uniform subhypergraph of 𝐻includes two hyperedges of cardinality 3: 𝑒2={𝑢1,𝑢4,𝑢5}and𝑒5={𝑢2,𝑢5,𝑢6}, with
their intersection being the vertex 𝑢5.
A mobility flow network is a weighted directed graph where nodes represent locations and directed edges
indicate the movement of individuals from one node to another, and edge weights quantify the magnitude of
that movement. However, this representation cannot accurately capture complex flows, namely, movements of
individuals among more than two locations (see Figure 6 in Appendix).
Colocation maps [ 8] estimate the likelihood that individuals from different home locations are present in the
same place at the same time, as illustrated by a colocation event in Figure 6B in Appendix. But they do not
have information about where the colocations occur. These maps reveal how populations from various locations
come into contact, emphasizing the spatial and temporal overlap of individuals within a given time window.
However, this approach is limited to pairwise relationships between locations and does not account for the
dynamic interactions or movement flows between them.
Chang et al. 2021 [ 17] proposed a bipartite network representation where neighbourhoods (census block groups)
form one set of nodes and point-of-interests (POIs, such as restaurants, shopping centers) form another set of
, Vol. 1, No. 1, Article . Publication date: March 2025.
Identifying and Characterising Higher Order Interactions in Mobility Networks Using Hypergraphs • 5
nodes. They incorporate this representation in epidemic modelling to predict inequities faced among different
groups of people during COVID-19.
All the above existing location-to-location mobility data representations solely focus on pairwise interactions.
3 Methods
In this section, we present steps in our methodology for constructing a co-visitation hypergraph from individual-
level mobility data. Figure 2 provides a schematic overview of the key steps in the process formalized in the
pseudocode of the Algorithm 1.
3.1 Data Preprocessing and Notations
Let the region of interest be partitioned into a set of locations L={ℓ1,ℓ2,...,ℓ𝑛}that collectively cover the entire
geographical region of interest. Define [𝐷]={1,2,...,𝐷}as the set of days and U={𝑢1,...,𝑢𝑟}as the set of
individuals. The mobility trajectories are represented by a function D:U×[𝐷]→ 2L, mapping a pair of an
individual and a day to a subset of locations in L. Specifically, D(𝑢𝑖,𝑑)denotes the set of locations visited by
individual𝑢𝑖on day𝑑, disregarding the sequence and frequency of visits to each location.
We pre-process the individual-level mobility data to construct D. Figure 2A provides an example of sample
trajectories for three individuals, 𝑢1,𝑢2, and𝑢3, shown in both raw format and the resulting set-based notation.
While the unit of time is assumed to be one day in this section, this can be adjusted by redefining Daccordingly.
3.2 Observation Windows
To identify sets of locations frequently co-visited by individuals, we segment time into overlapping observation
windows of length Δ𝑇. These windows are generated by applying a sliding window of length Δ𝑇over the𝐷days,
resulting in a sequence of observation windows {𝑤1,...,𝑤𝐷−Δ𝑇+1}. The𝑡𝑡ℎobservation window, 𝑤𝑡, encompasses
the days{𝑡,𝑡+1,...,𝑡+Δ𝑇−1}(see Figure 2B).
For each observation window, we aggregate the set of locations visited by an individual during the window,
ignoring the frequency and order of visits. This approach ensures we focus on co-visited locations without
considering temporal details within the window. Figure 2B illustrates the concept of observation windows.
3.3 Identifying Frequently Co-visited Locations Using Frequent Pattern Mining
We use frequent pattern mining techniques [ 18,19] to identify sets of locations frequently co-visited by individuals
in the mobility data. Frequent pattern mining involves discovering frequent itemsets in a transactional database,
where each transaction consists of a subset of all items.
To apply this methodology, we first prepare a transactional dataset. For an individual 𝑖, the set of locations visited
during an observation window 𝑤𝑡={𝑡,···,𝑡+Δ𝑇−1}is defined as:
𝑇𝑗=𝑡+Δ𝑇−1Ø
𝑑=𝑡D(𝑖,𝑑),
where D(𝑖,𝑑)represents the set of locations visited by individual 𝑖on day𝑑.
We treat each 𝑇𝑗as a single transaction. Thus, for each individual and each observation window, we create one
transaction, resulting in up to 𝑀≤(𝐷−Δ𝑇+1)𝑟transactions, where 𝐷is the total number of days, Δ𝑇is the
window size, and 𝑟is the number of individuals. The resulting transaction dataset is denoted as T={𝑇1,···,𝑇𝑀}.
Figure 2C shows an example of such a transaction dataset.
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6 • Prathyush Sambaturu, Bernardo Gutierrez, and Moritz U.G. Kraemer
`1`2`3`4`5`6`7`8`9u1u2
u3l1l2l5l4l6l7l8ASample trajectories a three individuals on a day 1 
1234…D-2D-1DDays`1`2`3`4`5`6`7`8`9
`1`2`3`4`5`6`7`8`9`1`2`3`4`5`6`7`8`9
𝑢!𝑢"𝑢#𝑤!𝑤"𝑤#$"BIndividual visits in each time window for Δ𝑇	=	3Items: Set of locations 𝑙!,𝑙",𝑙%,𝑙&,𝑙',𝑙(,𝑙),𝑙*,𝑙+Transactions: Set of locations visited by an individual within a specified time window.Transaction dataset below has a total of M transactions including of  other individuals and time windows.CConstructing Transaction Dataset 
DEIdentifying Frequently Co-Visited Locations using the FPGrowth AlgorithmT. No.U. IDWindowTransactionT1u1w1{l1,l2,l4,l5,l6}T2u2w1{l4,l5,l6,l7}T3u3w1{l4,l5,l8}............Tku1w2{l1,l2,l3,l5}Tk+1u2w2{l5,l6,l7}Tk+2u3w2{l4,l5,l6,l8}............Tru1wD 2{l1,l2,l5}Tr+1u2wD 2{l4,l5,l6,l7,l8}Tr+2u3wD 2{l4,l7,l8,l9}............TM.........
Hypergraph ConstructionNode set: Set of all locationsEdge set: Frequently co-visited locations/Frequent itemsets of size ≥2InputT={T1,T2,···,TM}minsup= 0.05 (5%)minsize=2FPGrowthFrequent Itemsets of size 2ItemsetSupport{`4,`5}0.14 (14%){`5,`7}0.125 (12.5%){`5,`6}0.12 (12%){`2,`5}0.11 (11%){`3,`5}0.09 (9%){`2,`4}0.083 (8.3%){`4,`5,`7}0.072 (7.2%){`7,`8}0.065 (6.5%){`4,`5,`6}0.059 (5.9%){`8,`9}0.054 (5.4%){`1,`2}0.051 (5.1%)
Fig. 2. Schematic of constructing a co-visitation hypergraph from individual-level human movement data. A) (left)
Sample trajectories for three individuals, 𝑢1,𝑢2, and𝑢3, 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: {ℓ1,ℓ2,ℓ4,ℓ5}for𝑢1,{ℓ4,ℓ5,ℓ6,ℓ7}for
𝑢2, and{ℓ4,ℓ8}for𝑢3.B) A sliding time window of length Δ𝑇=3days moves across a sequence of 𝐷days, creating
observation windows 𝑤1,𝑤2,...,𝑤𝐷−2. For representative windows 𝑤1,𝑤2shown, and 𝑤𝐷−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
𝑢1, red for𝑢2, and green for 𝑢3).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
T={𝑇1,···,𝑇𝑀}, allowing the identification of frequently co-visited locations across 𝑀individuals over 𝐷days.
Potential candidates for frequently co-visited locations (frequent itemsets) are highlighted: {ℓ1,ℓ2}and{ℓ4,ℓ5,ℓ6}.D)
The FPGrowth algorithm takes the transaction dataset 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 ℓ5has the highest degree (7),
while locations ℓ1,ℓ3, andℓ9have the lowest degree (1). Both hyperedges of size 3 are incident on ℓ5.
Anitemset , denoted by 𝑒⊆L, is a subset of locations (items) that can appear in any transaction in the dataset.
The support of an itemset 𝑒, denoted by S(𝑒), is the proportion of transactions in Tthat contain 𝑒. An itemset
is considered frequent if its support exceeds a user-defined threshold, 𝑚𝑖𝑛_𝑠𝑢𝑝, which specifies the minimum
required proportion of transactions containing the itemset. For example, setting 𝑚𝑖𝑛_𝑠𝑢𝑝=0.05means an itemset
is frequent if it appears in at least 5%of the total transactions, or equivalently, in at least 0.05𝑀transactions.
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Identifying and Characterising Higher Order Interactions in Mobility Networks Using Hypergraphs • 7
A frequent itemset 𝑒⊆L is called a maximal frequent itemset if no superset 𝑒′⊃𝑒is frequent. In other words, 𝑒
is frequent and adding any additional items to 𝑒results in an itemset that does not meet the minimum support
threshold.
We apply the FP-Growth algorithm [ 19] to compute frequent itemsets from our transaction dataset T. The
algorithm requires two inputs: the minimum support threshold, 𝑚𝑖𝑛_𝑠𝑢𝑝, and the minimum size of an itemset,
𝑚𝑖𝑛_𝑠𝑖𝑧𝑒. FP-Growth first constructs a compact FP-tree (Frequent Pattern Tree) from the transaction data, encoding
the frequency of itemsets while preserving their hierarchical relationships. It then recursively mines the FP-tree
to extract frequent location patterns Fand their corresponding supports S.
In this context, the frequent itemsets represent sets of locations co-visited by individuals with high support within
observation windows of length Δ𝑇. These patterns provide insights into mobility behaviors in the population
and highlight significant co-visitation trends.
3.4 Constructing the Co-visitation Hypergraph
We define the co-visitation hypergraph as H=(𝑉,E,𝑤), where𝑉=Lrepresents the set of locations, Edenotes
the set of hyperedges, and 𝑤:E→Ris a weight function that assigns a weight 𝑤(𝑒)to each hyperedge 𝑒∈E.
For each frequent itemset 𝑒∈F, a corresponding hyperedge is added to E. The weight of a hyperedge 𝑒is defined
as the support of the corresponding itemset 𝑒, given by𝑤(𝑒)=S(𝑒).
The resulting hypergraph H(example in Figure 2E) encodes the relationships among frequently co-visited
locations, with the weights of the hyperedges reflecting the frequency of co-visitation.
Algorithm 1 Constructing a Co-Visitation Hypergraph from Mobility Data
Require: LocationsL, Mobility data Dfor𝑟individuals over 𝐷days, Time window Δ𝑇,𝑚𝑖𝑛_𝑠𝑢𝑝,𝑚𝑖𝑛_𝑠𝑖𝑧𝑒
Ensure: Hypergraph H=(L,E,𝑤)
1:InitializeT←∅ ,H←(L,∅) ⊲Initialize dataset and hypergraph
2:foreach window 𝑤𝑡of length Δ𝑇over𝐷days do ⊲Slide observation window
3: foreach individual 𝑖∈{1,...,𝑟}do ⊲Process movements for each individual
4: Compute𝑇𝑗=Ð𝑡+Δ𝑇−1
𝑑=𝑡D(𝑖,𝑑) ⊲Group locations visited by individual in window
5: if𝑇𝑗≠∅then add𝑇𝑗toT ⊲Add non-empty transaction
6: end if
7: end for
8:end for
9:F,S←FPGrowth(T,𝑚𝑖𝑛 _𝑠𝑢𝑝,𝑚𝑖𝑛 _𝑠𝑖𝑧𝑒) ⊲Mine frequent itemsets and supports
10:foreach𝑒∈Fdo ⊲Construct hypergraph edges with support as edge weights
11: Add hyperedge 𝑒toEand set𝑤(𝑒)← S(𝑒)
12:end for
13:return H ⊲Return the constructed hypergraph
4 Results
4.1 Dataset Overview and Experiment Setup
This section provides an overview of the datasets used in our study and outlines the experimental setup.
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8 • Prathyush Sambaturu, Bernardo Gutierrez, and Moritz U.G. Kraemer
4.1.1 Datasets. We use two public, city-scale, individual-level human mobility datasets [ 16] to construct
mobility hypergraphs, analyze their structural properties, and investigate mobility patterns. Both datasets contain
movement records over 75 days with fields: user ID ( uid), day (0 to 74), timeslot (30-minute intervals labeled 0 to
47), and x/y coordinates mapped to a 200x200 grid, where each grid cell covers a 500-meter square area.
The first dataset, DS1, captures the movements of 100,000 individuals over 75 days. The second dataset, DS2,
includes movements of 25,000 individuals during 60 regular days, followed by 15 days during an emergency
scenario. No information about the nature of the emergency scenario is provided. In addition, the data sets
provide the number of points of interest (POIs) by category for each location. Detailed dataset preparation can be
found in Yabe et al. 2024 [20].
4.1.2 Selection of spatial resolution and parameters. The application of our method to the above datasets
involves making choices regarding spatial resolution and parameter selection ( Δ𝑇,min_sup ) as outlined below:
(1)Spatial Resolution : The original 200x200 grid is aggregated into a 20x20 grid using a scaling factor of 10,
where each cell represents a 25-km2area.
(2)Temporal Window ( Δ𝑇): We examine three temporal resolutions: Δ𝑇∈{1,3,7}days, corresponding to
daily, three-day, and weekly observation windows.
(3)Support Threshold ( min_sup ): We explore support thresholds min_sup∈{0.005,0.01,0.015}, corresponding
to 0.5%, 1%, and 1.5% minimum supports in the transactional dataset.
Table 1. Experimental Parameters and Their Descriptions
Parameter Values Description
Spatial scaling factor 10 Aggregates the grid to 20x20, where each cell covers 25 km2
Temporal window size ( Δ𝑇){1,3,7}days Daily, three-day, and weekly observation windows
Support threshold ( min_sup ){0.005, 0.01, 0.015} Minimum support thresholds: 0.5%, 1%, and 1.5%
Table 1 summarizes the experimental parameters. This design facilitates a comprehensive analysis of co-visitation
patterns and mobility hypergraph structures under varying temporal resolutions and support thresholds.
4.2 Impact of 𝑚𝑖𝑛_𝑠𝑢𝑝andΔ𝑇on the Spatial Spread of Co-Visitation Hypergraphs
We explore how the time window size ( Δ𝑇) and the minimum support threshold ( 𝑚𝑖𝑛_𝑠𝑢𝑝) influence the structural
and spatial characteristics of co-visitation hypergraphs. Figures 7A-B (in Appendix) illustrate co-visitation
hypergraphs for daily ( Δ𝑇=1) and three-day ( Δ𝑇=3) observation windows, respectively. For a fixed 𝑚𝑖𝑛_𝑠𝑢𝑝
threshold, increasing Δ𝑇results in a significant increase in the number and spatial coverage of hyperedges
reflecting how a longer temporal window captures a more diverse set of interactions across the spatial grid.
Figures 7C-D (in Appendix) provide visualizations of 5-uniform subhypergraphs derived from the co-visitation
hypergraphs for Δ𝑇=3andΔ𝑇=7, respectively, with 𝑚𝑖𝑛_𝑠𝑢𝑝=0.005.
We compute maximum chebyshev distance between nodes in each hyperedge to examine the spatial proximity of
higher-order interactions. The chebyshev distance[ 21] between two nodes at coordinates (𝑥1,𝑦1)and(𝑥2,𝑦2)
is defined as 𝑑∞((𝑥1,𝑦1),(𝑥2,𝑦2))=max(|𝑥1−𝑥2|,|𝑦1−𝑦2|). For each hyperedge 𝑒, the maximum chebyshev
distance is given by 𝑑∞(𝑒)=max𝑢,𝑣∈𝑒𝑑∞(𝑢,𝑣), where𝑢and𝑣are any two nodes in the hyperedge 𝑒. Therefore,
the maximum chebyshev distance in a hypergraph 𝐻as𝐷∞(𝐻)=max𝑒∈E𝑑∞(𝑒). ForΔ𝑇=3, the maximum
chebyshev distance in the 5-uniform subhypergraph is 3, while for Δ𝑇=7, it increases to 4. This demonstrates
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Identifying and Characterising Higher Order Interactions in Mobility Networks Using Hypergraphs • 9
how larger temporal windows not only lead to more higher-order interactions but also allow interactions to
occur over greater spatial distances.
Visualising the hypergraphs, we observe that increasing Δ𝑇leads to a more diffuse structure, capturing interactions
that span broader spatial regions. Simultaneously, varying 𝑚𝑖𝑛_𝑠𝑢𝑝alters the density and strength of interactions
within the hypergraph, with lower thresholds enabling a richer representation of co-visitation patterns. These
findings suggests how the choice of parameter will reflect in the emergence of distinct patterns in the co-visitation
hypergraph.
4.3 Structural and Spatial Properties of the Co-visitation Hypergraph
This section analyzes the structural and spatial characteristics of the co-visitation hypergraph by examining
degree distributions, hyperedge sizes, spatial arrangements, and higher-order interactions.
To characterize the degree distribution, we examine the complementary cumulative distribution function
(CCDF)[ 22] of node degrees in a hypergraph H=(𝑉,E), denoted as 𝑃(Degree≥𝑘), which represents the
probability that a randomly selected node has a degree of at least 𝑘defined as below:
𝑃(Degree≥𝑘)=|{𝑣∈𝑉:Degree(𝑣)≥𝑘}|
|𝑉|,
where|𝑉|is the total number of nodes in the hypergraph.
Figure 3A shows the CCDF of node degrees for hypergraphs with 𝑚𝑖𝑛_𝑠𝑢𝑝=0.005andΔ𝑇∈{1,3,7}. While the
curves exhibit a heavy tail, the degree distribution is better modeled by an exponential distribution rather than a
power-law. Statistical fitting yielded log-likelihood ratios favoring an exponential distribution over a power-law
for all Δ𝑇values, with the best fit given by:
𝑃(𝑘)∝𝑒−𝜆𝑘,
where𝜆is the decay rate.
AsΔ𝑇increases, the decay rate 𝜆decreases, reflecting a broader degree distribution. This means that longer
observation windows ( Δ𝑇=7) capture more medium- and high-degree nodes than shorter windows ( Δ𝑇=1).
This transition from localized interactions to broader connectivity aligns with the increasing temporal scope,
which aggregates more co-visitation patterns.
Figure 3B illustrates the hyperedge size distribution across Δ𝑇∈{1,3,7}. The hyperedge size, defined as the
number of nodes in a hyperedge, is plotted on the x-axis, with the y-axis showing the frequency in log-scale. The
maximum hyperedge size increases from 4 for Δ𝑇=1to 5 for Δ𝑇=3, and 7 for Δ𝑇=7, emphasizing the growth
of higher-order interactions as Δ𝑇increases. Notably, larger hyperedges are more prevalent in weekly windows
(Δ𝑇=7), indicating that longer temporal windows facilitate higher-order interactions.
Figure 3C examines the maximum chebyshev distance among nodes in hyperedges of size at least 3 (i.e., only in
higher-order interactions). The plot shows that for 𝑚𝑖𝑛_𝑠𝑢𝑝=0.005, the maximum distance increases sharply
fromΔ𝑇=3toΔ𝑇=7, indicating more spatially distributed interactions. For higher 𝑚𝑖𝑛_𝑠𝑢𝑝values, the increase
is greater from Δ𝑇=1toΔ𝑇=3, reflecting a reduced presence of high-support interactions occurring over large
distances.
Figure 3D-F displays spatial heatmaps of node degrees on a 20x20 grid for 𝑚𝑖𝑛_𝑠𝑢𝑝=0.005andΔ𝑇∈{1,3,7}.
Each node’s position corresponds to its (𝑥,𝑦)coordinates, with the intensity of the red shading indicating its
degree. As Δ𝑇increases from 1 to 7, the spatial distribution of hyperedges becomes more diffuse, covering larger
regions of the grid. The maximum node degree increases significantly, from 44 for Δ𝑇=1, to 502 for Δ𝑇=3, and
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10 • Prathyush Sambaturu, Bernardo Gutierrez, and Moritz U.G. Kraemer
AB
DEFSpatial Heatmap of Node Degrees
C
Fig. 3. Structural and Spatial Properties of the Co-Visitation Hypergraph. A)CCDF of node degrees for hypergraphs
with𝑚𝑖𝑛_𝑠𝑢𝑝=0.005andΔ𝑇∈{1,3,7}(green, blue, and red, respectively). The x-axis shows node degree, and the
y-axis shows 𝑃(Degree≥𝑘).B)Hyperedge size distribution for Δ𝑇∈{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 Δ𝑇=1, and 5 and 7
forΔ𝑇=3andΔ𝑇=7.C)Maximum chebyshev distance over Δ𝑇values. X-axis represents Δ𝑇and y-axis represents
the max chebyshev distance. Each curve corresponds to a particular 𝑚𝑖𝑛_𝑠𝑢𝑝value. D,E,F) Spatial heatmaps of
node degrees for hypergraphs with 𝑚𝑖𝑛_𝑠𝑢𝑝=0.005andΔ𝑇∈{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.
to 7402 for Δ𝑇=7, reflecting an increasing aggregation of nodes into hyperedges over longer temporal windows.
Central high-degree nodes contribute to more hyperedges as Δ𝑇increases, emphasizing their role in broader
co-visitation patterns. Given the city-scale analysis, few locations are naturally highly connected, as reflected in
the heatmaps.
4.4 Structural and Spatial Comparisons of Co-Visitation Patterns Between Regular and
Emergency Days
Figure 4A presents the hyperedge size distribution for regular and emergency days at 𝑚𝑖𝑛_𝑠𝑢𝑝=0.005. The x-axis
represents hyperedge size, while the y-axis, plotted on a logarithmic scale, shows the frequency of hyperedges of
that size. Regular-day hypergraphs (blue curves) exhibit a higher frequency of larger hyperedges compared to
emergency-day hypergraphs (red curves). This difference is particularly pronounced for larger Δ𝑇values, such
as 7, where the longer observation window allows for greater aggregation of co-visitation patterns.
The maximum Chebyshev distance computed over all hyperedges of size ≥3analyzed in Figure 4B captures the
largest spatial extent of higher-order interactions in a hypergraph. For regular days (solid bars) and emergency
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Identifying and Characterising Higher Order Interactions in Mobility Networks Using Hypergraphs • 11
days (striped bars), the chebyshev distance increases with Δ𝑇and decreases as 𝑚𝑖𝑛_𝑠𝑢𝑝increases. The largest
differences between regular and emergency days are observed at 𝑚𝑖𝑛_𝑠𝑢𝑝=0.005, where the chebyshev distance
during emergencies is smaller by 1 consistently over all time window sizes. This reduction highlights the
restricted spatial spread of higher-order interactions during emergencies, consistent with localized mobility
patterns observed in Figure 4A.
Figures 4C visualizes the unique edges in co-degree graphs for regular and emergency hypergraphs. A co-degree
graph represents the relationships between node pairs based on their co-occurrence in hyperedges. Formally,
the co-degree of a pair of nodes (𝑢,𝑣)is the number of hyperedges in which both nodes appear together [ 23].
The edge weight in a co-degree graph corresponds to this co-degree value, effectively summarizing pairwise
interaction strength. We particularly restrict our co-degree graphs to focus on the hyperedges of size ≥3, thereby
restricting the analysis to higher-order interactions. In Figures 4C-D, blue edges represent co-degree relationships
unique to regular days, while red edges indicate those unique to emergency days. The thickness of each edge
corresponds to its co-degree value, with thicker edges indicating stronger pairwise interactions. Edges common
to both regular and emergency days are excluded from the plots. As Δ𝑇increases to 7 (Figure 4D), the number of
unique edges increases, extending over a broader spatial range. This analysis provides insights into locations
that are co-visited exclusively during emergencies or remain popular during regular days but see reduced or no
co-visitation in emergency scenarios.
The hyperedge size distributions and co-degree graph visualizations underscore key differences in mobility
patterns between regular and emergency days. Regular-day hypergraphs exhibit broader spatial coverage, more
higher-order interactions, particularly for larger Δ𝑇.
4.5 Analysis of POI Distribution and its Relationship with Node Degree in Hypergraphs
The spatial distribution of Points of Interest (POI) plays a crucial role in shaping mobility patterns, as locations
with higher POI density have higher gravity and often attract greater foot traffic [ 24]. Figure 5 provides insights
into the distribution of POIs across the spatial grid, their clustering patterns, and their relationship with degree
centrality in co-visitation hypergraphs.
Spatial Distribution of POIs. The heatmap in Figure 5A reveals significant heterogeneity in the distribution of
POIs across the 20x20 grid. A few locations exhibit a markedly higher density of POIs, as indicated by darker
shades of red, suggesting that these areas may serve as hubs of activity within the city. These nodes contribute
disproportionately to the overall mobility network by attracting individuals for various purposes, ranging from
work to leisure.
Clustering of Locations Based on POI Categories. To explore patterns in POI composition, Figure 5B presents
clusters of locations derived using K-means clustering. Each location is represented as a vector containing the
counts of POIs in various categories (e.g., retail, healthcare, and education). By using the similarity of these
POI vectors as the clustering metric, three distinct clusters are identified, with the optimal number of clusters
determined via the Elbow method and kneedle detection[ 25] to minimize Within-Cluster Sum of Squares (WCSS).
We find that the optimal number of clusters is just 3. A limitation to finding more nuanced clustering is the fact
that the categories are anonymized in this dataset for privacy purposes.
Relationship Between POI Density and Node Degree. Figure 5C examines the correlation between the total POI
count at each location and its degree in hypergraphs constructed for varying temporal window lengths ( Δ𝑇) with
𝑚𝑖𝑛_𝑠𝑢𝑝=0.015. Degree centrality represents the total number of hyperedges incident on a node, capturing the
extent of co-visitation interactions associated with the location. The scatter plot indicates a positive correlation
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12 • Prathyush Sambaturu, Bernardo Gutierrez, and Moritz U.G. Kraemer
A
CB
D
Fig. 4. Comparison of hypergraphs capturing mobility patterns on regular and emergency days. A) Hyperedge
size distribution for Δ𝑇values at𝑚𝑖𝑛_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 Δ𝑇. Larger
hyperedges are more frequent on regular days, with differences increasing at higher Δ𝑇.B) Maximum Chebyshev
distance for higher-order interactions (hyperedges ≥3) across Δ𝑇and𝑚𝑖𝑛_𝑠𝑢𝑝. Solid bars show regular days, striped
bars emergency days, with colors for 𝑚𝑖𝑛_𝑠𝑢𝑝values. 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 Δ𝑇=3,𝑚𝑖𝑛_𝑠𝑢𝑝=0.005andΔ𝑇=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.
between POI count and node degree, suggesting that locations with higher POI density tend to play a more
central role in the co-visitation network.
The best-fit parameters for the power-law model of the form 𝑦=𝑎·𝑥𝑏, where𝑦represents the node degree and 𝑥
represents the total POI count, are estimated. The dashed black line in Figure 5C visualizes the fit. The observed
power-law relationship highlights the disproportionate influence of high-POI locations, which act as hubs in the
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Identifying and Characterising Higher Order Interactions in Mobility Networks Using Hypergraphs • 13
Correlation Between Node Degree in Co -Visitation Hypergraphs and POI Count Across Temporal Window LengthsA B
C
Fig. 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 ( Δ𝑇). The x-axis represents the degree of a location in hypergraph
for given Δ𝑇and𝑚𝑖𝑛_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.
mobility network. Overall, this analysis underscores the strong interplay between urban structure, as reflected by
POI distributions, and the structural properties of mobility hypergraphs.
5 Discussion
In this paper, we applied our approach to a city-scale mobility dataset and revealed the structure and spatial
properties of co-visitation hypergraphs in this specific context. However, our methods could be applied to inter-
city or regional mobility data to reveal broader co-visitation patterns. Further, the comparison of flow networks
with co-visitation hypergraphs has potential to highlight scenarios where hypergraphs offer deeper insights.
Applying this approach on longer temporal datasets could enhance understanding of seasonal and long-term
mobility trends.
5.1 Potential Applications
The co-visitation hypergraph framework is valuable for analyzing group-level interactions, particularly in
applications like contact tracing in infectious disease epidemiology.
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14 • Prathyush Sambaturu, Bernardo Gutierrez, and Moritz U.G. Kraemer
During epidemics, hypergraphs can identify higher-order movement patterns driving disease spread, supporting
targeted interventions like optimized testing or localized travel restrictions. Contact tracing would normally
collect the information about places an infected individual visited, but the hypergraph approach allows to estimate
the plausible reach of individuals at these locations. This reach can be tailored to epidemiologically meaningful
frameworks (i.e., using a Δ𝑇concordant with infection progression). For contact tracing, analyzing mobility
within a defined time window (e.g., three days) helps track all locations visited by exposed individuals. If infections
emerge at a site, co-visitation hypergraphs help reveal all connected locations within the time window, facilitating
efficient surveillance and containment.
In climate emergencies, hypergraphs capture shifts in mobility behaviors during crises. Certain movement
patterns (hyperedges) may emerge or disappear, which traditional mobility networks often miss. By offering
a more comprehensive view of movement adaptations, hypergraphs enhance evacuation planning, resource
allocation, and post-disaster recovery by identifying critical travel routes and behavioral shifts.
5.2 Limitations of the Current Study
Our approach to constructing mobility hypergraphs relies on individual-level mobility data, which limits its
applicability to datasets with such granularity. The datasets used in our study span only 75 days, with emergency
scenarios covering just 15 days in DS2. Longer time-series data would allow for a more comprehensive study
of temporal evolution in co-visitation patterns, particularly during emergencies. Furthermore, the datasets
anonymize location names and Points of Interest (POIs), which restricts the ability to conduct application-
oriented analyses such as understanding the role of specific types of locations (e.g., schools, hospitals, markets)
in mobility dynamics. Finally, our study is constrained to city-scale data; incorporating datasets with larger
spatial coverage, such as inter-city or regional mobility, could provide insights into how structural properties of
hypergraphs vary across broader geographic scales.
6 Conclusions
We introduce a novel representation of mobility data that captures higher-order interactions among locations. We
propose a method for constructing co-visitation hypergraphs using individual-level mobility trajectory data. Our
results validate the utility of hypergraphs in capturing complex, higher-order mobility patterns, making them
valuable for urban mobility analysis, public health, and disaster management. In conclusion, our hypergraph-based
mobility analysis framework advances research in urban planning, public health, and disaster resilience, opening
new avenues for future applications.
Data, Materials, and Software Availability
All code used in this work is openly available from GitHub at https://github.com/prathyushsambaturu/Covisitation-
hypergraphs.git. Data used in this work are publicly available from Yabe et al. 2024 [16].
Acknowledgments
M.U.G.K. acknowledges funding from The Rockefeller Foundation (PC-2022-POP-005), Google.org, the Oxford
Martin School Programmes in Pandemic Genomics (& B.G.) & Digital Pandemic Preparedness, European Union’s
Horizon Europe programme projects MOOD (#874850) and E4Warning (#101086640), Wellcome Trust grants
303666/Z/23/Z, 226052/Z/22/Z (& B.G.) & 228186/Z/23/Z, the United Kingdom Research and Innovation (#APP8583),
the Medical Research Foundation (MRF-RG-ICCH-2022-100069), UK International Development (301542-403), the
Bill & Melinda Gates Foundation (INV-063472) and Novo Nordisk Foundation (NNF24OC0094346). The contents of
this publication are the sole responsibility of the authors and do not necessarily reflect the views of the European
Commission or the other funders.
, Vol. 1, No. 1, Article . Publication date: March 2025.
Identifying and Characterising Higher Order Interactions in Mobility Networks Using Hypergraphs • 15
Author contributions
P.S., B.G., and M.U.G.K. conceptualised the study. P.S. performed the research and analysed the data. M.U.G.K.
supervised the work. P.S. wrote the initial manuscript with critical input from M.U.G.K. and B.G. All authors
edited and revised the manuscript. M.U.G.K. acquired funding for the research.
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Identifying and Characterising Higher Order Interactions in Mobility Networks Using Hypergraphs • 17
Appendix
7 Limitations of Existing Location-to-Location Interactions Based Representations of Human
Mobility
A mobility flow network 𝐺=(𝑉,𝐸)is a weighted directed graph where nodes in 𝑉represent locations, edges
in𝐸indicate movement of individuals between location pairs, and edge weights quantify the intensity of that
movement. For example, consider the flow between locations (ℓ𝑖,ℓ𝑘)∉𝐸(locations without a direct edge in the
flow network 𝐺), we need to consider the paths connecting them. For instance, if 𝑃={ℓ𝑖,ℓ𝑗,ℓ𝑘}is a path in 𝐺
where(ℓ𝑖,ℓ𝑗)and(ℓ𝑗,ℓ𝑘)are edges in 𝐸, this indicates movement between ℓ𝑖andℓ𝑗, as well as between ℓ𝑗andℓ𝑘.
However, this does not guarantee that some individuals actually traveled in the exact sequence ℓ𝑖→ℓ𝑗→ℓ𝑘or
even that they visited all three locations together in a time interval. In other words, the existence of such a path
does not imply that any person followed that specific route. This is illustrated in Figure 6A. The mobility flow
network representation also overlooks complex interactions, like flow that includes more than two locations,
which could offer more detailed insights into human mobility patterns.
Fig. 6. Location-to-Locations Interactions in mobility flow networks and colocation maps. A) The pairwise interactions
in a flow network, (ℓ𝑖,ℓ𝑗)and(ℓ𝑗,ℓ𝑘), do not confirm the existence of a path ℓ𝑖→ℓ𝑗→ℓ𝑘(shown in red dashed
lines) traveled by individuals within the same time interval (e.g., on the same day). Additionally, they do not indicate
whether the locations {ℓ𝑖,ℓ𝑗,ℓ𝑘}will be visited together on a single day. B) The figure shows a colocation event where
individuals from locations ℓ𝑖andℓ𝑗are in the same place at the same time, captured as a pairwise interaction in
colocation maps.
Colocation maps estimate the likelihood that individuals from different home locations are present in the same
place at the same time, as illustrated by a colocation event in Figure 6B. These maps reveal how populations
from various locations come into contact, emphasizing the spatial and temporal overlap of individuals within a
given time window. However, this approach is limited to pairwise relationships between locations and does not
account for the dynamic interactions or movement flows between them.
In this work, we focus on higher-order flows among locations, capturing frequent mobility patterns within a
region. Humans visit sequences of locations within a given time window (e.g. a day or during a week, weekend),
forming trajectories on a map. For example, we examine the proportion of individuals in a population visiting
three locations— ℓ𝑖,ℓ𝑗, andℓ𝑘—in any order during a specified period.
Building on this concept, we generalize mobility networks by representing frequent mobility patterns as higher-
order interactions between locations, encoded as hyperedges in a mobility hypergraph. This hypergraph-based
approach offers a versatile framework for analyzing and applying mobility flows in various contexts.
, Vol. 1, No. 1, Article . Publication date: March 2025.
18 • Prathyush Sambaturu, Bernardo Gutierrez, and Moritz U.G. Kraemer
Unlike pairwise centrality in traditional networks, hypergraph centrality can identify not just key nodes (locations)
but also pivotal hyperedges (groups of locations) that are central to movement patterns. This is particularly
useful for pinpointing influential areas in disease spread or critical hubs during emergencies, providing actionable
insights for intervention. By integrating spatial, temporal, and structural dimensions, mobility hypergraphs offer
a powerful tool for comprehensive analysis and informed decision-making.
8 Undirected vs. Directed Mobility Hypergraphs
In the mobility hypergraph H, each hyperedge represents an interaction between two or more locations. The
edges are typically undirected, as in many applications, the sequence of visits to these locations may not be
relevant. For instance, in contact tracing for epidemic control, it is often sufficient to identify the set of locations
an individual has visited, regardless of the order in which those visits occurred. However, in other contexts, the
order of location visits may have significant importance. In such cases, the same approach can be extended to
construct a directed hypergraph[ 26], where the hyperedges (hyperarcs) represent ordered sequences of locations
rather than unordered sets.
9 Impact of 𝑚𝑖𝑛_𝑠𝑢𝑝andΔ𝑇on the Evolution of Co-Visitation Hypergraphs
We explore how the time window size ( Δ𝑇) and the minimum support threshold ( 𝑚𝑖𝑛_𝑠𝑢𝑝) influence the evolution
of co-visitation hypergraphs, focusing on their structural and spatial characteristics. Figures 7A-B illustrate
co-visitation hypergraphs for daily ( Δ𝑇=1) and three-day ( Δ𝑇=3) observation windows, respectively. Nodes
are positioned on a 20x20 spatial grid based on their coordinates, and hyperedges are visualized as elastic
bands connecting incident nodes. The hyperedge colors represent their support levels: green ( [0.005,0.01)), red
([0.01,0.015)), and blue ([0.015,∞)). For a fixed 𝑚𝑖𝑛_𝑠𝑢𝑝threshold, increasing Δ𝑇results in a significant rise
in the number and spatial coverage of hyperedges. For example, for 𝑚𝑖𝑛_𝑠𝑢𝑝=0.01(corresponding to 1%), the
hyperedges (shown in blue) expand from being concentrated around high-degree nodes for Δ𝑇=1to a larger
spatial distribution for Δ𝑇=3. This reflects how a longer temporal window captures a more diverse set of
interactions across the spatial grid.
Figures 7C-D provide visualizations of 5-uniform subhypergraphs derived from the co-visitation hypergraphs
forΔ𝑇=3andΔ𝑇=7, respectively, with 𝑚𝑖𝑛_𝑠𝑢𝑝=0.005. These subhypergraphs contain only hyperedges of
size 5. The maximum Chebyshev distance between nodes in each hyperedge is calculated to examine the spatial
proximity of higher-order interactions. This is applicable in our context only because our nodes form grid cells in
a rectangular grid. The Chebyshev distance[ 21] between two nodes at coordinates (𝑥1,𝑦1)and(𝑥2,𝑦2)is defined
as:
𝑑∞((𝑥1,𝑦1),(𝑥2,𝑦2))=max(|𝑥1−𝑥2|,|𝑦1−𝑦2|).
For each hyperedge 𝑒, the maximum Chebyshev distance is given by:
𝑑∞(𝑒)=max
𝑢,𝑣∈𝑒𝑑∞(𝑢,𝑣),
where𝑢and𝑣are any two nodes in the hyperedge 𝑒. Therefore, we define a hypergraph H=(𝑉,E), the maximum
chebyshev distance is 𝐷∞(H)=max𝑒∈E𝑑∞(𝑒). ForΔ𝑇=3, the maximum Chebyshev distance in the 5-uniform
subhypergraph is 3, while for Δ𝑇=7, it increases to 4. This demonstrates how larger temporal windows not only
lead to more higher-order interactions but also allow interactions to occur over greater spatial distances.
, Vol. 1, No. 1, Article . Publication date: March 2025.
Identifying and Characterising Higher Order Interactions in Mobility Networks Using Hypergraphs • 19
AB
DC
Fig. 7. Visualization of Co-Visitation Hypergraphs and k-Uniform Subhypergraphs. A) Co-visitation hypergraph for
Δ𝑇=1and𝑚𝑖𝑛_𝑠𝑢𝑝=0.005, with nodes shown as black dots on a 20x20 spatial grid and hyperedges represented as
elastic bands. Hyperedge colors indicate support ranges: green for [0.005,0.01), red for[0.01,0.015), and blue for
[0.015,∞). The visualisations are generated using the HyperNetX package developed by Pacific Northwest National
Laboratory [ 27].B) Co-visitation hypergraph for Δ𝑇=3and𝑚𝑖𝑛_𝑠𝑢𝑝=0.01, depicted similarly to A, with hyperedge
colors limited to red and blue for corresponding support ranges. C-D) 5-uniform subhypergraphs of co-visitation
hypergraphs for Δ𝑇=3andΔ𝑇=7with𝑚𝑖𝑛_𝑠𝑢𝑝=0.005. Nodes, shown as black dots, are positioned on a cropped
spatial grid containing only subhypergraph nodes. All hyperedges are of size 5. The maximum Chebyshev distance
between connected nodes is 3 for Δ𝑇=3and 4 for Δ𝑇=7. These visualizations highlight variations in topology,
spatial extent, edge density, and spatial proximity of nodes in co-visitation hypergraphs with changes in Δ𝑇and
𝑚𝑖𝑛_𝑠𝑢𝑝.
, Vol. 1, No. 1, Article . Publication date: March 2025.