Filtering higher-order datasets

We present examples from social science, academia, and biology that illustrate how the scale of interactions can impact the underlying analysis.

2.1.1. Social networks

2.1.2. Email

2.1.3. Co-authorship networks

2.1.4. Protein networks

Consider a hypergraph $H = (V, E)$ where V is the set of nodes and E is the set of hyperedges, which are arbitrary nonempty subsets of the node set. We call an interaction between k entities a k-hyperedge, and a hypergraph which solely consists of k-hyperedges is called k-uniform. A simplicial complex is a special case of a hypergraph where the existence of a k-hyperedge—in this instance called a $(k-1)$-simplex—implies the existence of every possible subset of that hyperedge.

We now define the filtering of a hypergraph $H = (V,E)$ with respect to the filters $f: V \to \{0,1\}$ and $g: E \to \{0,1\}$ to be the hypergraph $H_{f,g} = (V_f,E_g)$ where

$$\begin{align*} V_f = \left\{v\in V \ | \ f\left(v\right) = 1\right\} \end{align*}$$

is the set of filtered nodes with respect to f and

$$\begin{align*} E_g = \left\{e\in E \ | \ g\left(e\right) = 1\right\}, \end{align*}$$

is the set of filtered hyperedges with respect to g. For simplicity, we denote the set of nodes induced by the filtered edges as $V_g = \{v \ | \ v\in e\in E_g\}$. We note that the choice of f must yield a node set V_f such that $V_g \subseteq V_f$. Three common choices for the filtered set of nodes are (1) $f(v) = 1$ and each filtering contains all the nodes, (2) $f(v) = \mathbb{I}_{V_f}(v)$, where $G_f = (V_f, E_f)$ is the giant component of the filtered edges E_f , and $\mathbb{I}$ is the indicator function, or (3), $f(v) = \mathbb{I}_{V_g}(v)$. In this paper, we solely consider the first choice to more easily compare nodal measures across different filterings. For ease of notation, we set $H_g \equiv H_{1,g}$.

We note that g may be chosen to reflect a variety of network properties, as well as metadata associated with hyperedges. Both pairwise networks and hypergraphs may have metadata, however, so the benefit of using higher-order datasets is the representation of multiple interaction sizes. In addition, unlike categorical attributes, filtering edges by their size allows us to define an ordering on the hypergraph filterings. These reasons motivate us to solely consider filters based on hyperedge size.

For a given non-negative integer k and a given comparison operator $*$, a size-dependent filtering $H_{(*,k)}$ is a filtering H_g where g is defined by

$$\begin{equation} g\left(e, k\right) = \begin{cases}1 &\mbox{if } |e| * k \\ 0 & \mbox{otherwise} \end{cases}. \end{equation} \tag{ 1 }$$

In the following, we define k as the filtering parameter.

More generally, one may define a size-dependent filtering based on a set K of hyperedge cardinalities, but in this work, we consider a single value k. Below, we present examples of common comparison operators.

• Uniform filtering, $H_{( = ,k)}$. This type of filtering may be used to isolate the structure associated with a particular hyperedge size.
• GEQ filtering, $H_{(\unicode{x2A7E}, k)}$. This filtering can be used to see the effect of excluding pairwise interactions or hyperedges with smaller cardinality. One can call this the 'higher-order' filtering for its ability to remove low-order interactions.
• LEQ filtering, $H_{(\unicode{x2A7D}, k)}$. This filtering can be used to exclude higher-order interactions. As before, one can also call this the 'lower-order filtering.' This has been used to construct some of the datasets in [53] (the author constructs hypergraphs with hyperedges of size $\unicode{x2A7D} 25$) and other papers that observe the effect of higher-order interactions by only including 2- and 3- hyperedges [54–56].
• Exclusion filtering, $H_{(\neq,m)}$. This filtering can be used to exclude interactions of a particular size. This can be helpful when considering datasets where a particular interaction size significantly alters the structure of a hypergraph—whether through a large number of edges or a significantly anomalous structure—and one desires to exclude that interaction size.

We illustrate these filtering types in figure 1. In principle, one can combine these operations to select or exclude more than one interaction size, but the filters presented in this paper form the basis for all other size-dependent filters.

Zoom In Zoom Out Reset image size

We can generate a set of filterings by considering different values of k or filtering operations. A set of filterings can either be disjoint or overlapping. A set of filterings, $\{H_{(*,k)} \ | \ k \in K\}$, where K is a set of interaction sizes, is disjoint if $E_{(*, i)}\cap E_{(*, j)} = \varnothing$, for all distinct $i,j\in K$. An example of a disjoint filtering set is $\{H_{( = ,k)} \ | \ k \in M\}$. A filtering set is overlapping when it is not disjoint. Examples of overlapping filtering sets are the complete sets of LEQ, GEQ, and exclusion filterings. Any hypergraph H can be expressed as the union of all its filterings—namely,

$$\begin{equation*} H = \left(\bigcup_{k\in K}V_{f_k},\bigcup_{k\in K}E_{f_k}\right). \end{equation*}$$

For the uniform filtering set, the edge-wise union is disjoint. When filtering simplicial complexes, only the LEQ filtering preserves the simplicial structure of the dataset, but one could, in principle, convert the simplicial complex to its equivalent hypergraph to allow the use of other filtering types.

It is common that despite the original dataset being fully connected, the filtered dataset is not, and we can deal with this in several ways.

First, we note that many higher-order measures are not dependent on the dataset being fully connected. For example, degree assortativity and modularity are aggregated over nodes and edges without any dependence on the connectedness of the dataset. Similarly, the degree of a node simply measures the number of hyperedges of which that node is a member.

This is not always the case, however. Several measures of centrality [20, 21] require the dataset to be connected. These eigenvector-based methods require a single connected component, although one can look at each connected component individually and determine the relative centrality of each node in that component. Alternatively, one can consider a regularized version of that metric, just as PageRank [5] is similar to eigenvector centrality and does not require the network to be connected. Likewise, in the case of community detection, when the dataset is not fully connected, there are several well-established ways to label isolated nodes and nodes not contained within the giant component [6].

Here, we briefly describe the structural measures we utilize in this study and refer the reader to appendix A for additional details.

Global measures. We consider the effective information (EI) and the degree assortativity as global measures of higher-order structure.

• EI is defined on the clique projection of the hypergraph as
  $$\begin{equation*} \textrm{EI} = \mathcal{H}\left(\langle W^\textrm{out}_i\rangle\right) - \langle \mathcal{H}\left(W^\textrm{out}_i\right)\rangle, \end{equation*}$$
  where $W^\textrm{out}_i$ is the $\ell_1$-normalized vector of out-degrees for node i and $\mathcal{H}$ is the Shannon entropy. EI measures the strength of unique associations in a network.
• Degree assortativity measures the tendency of nodes with similar degrees to connect with one another more often than would be expected at random. We utilized four different measures of degree assortativity; the first three—top-bottom, top-2, and uniform—defined as in [15], and the dynamical assortativity [16] defined as
  $$\begin{equation*} \rho = \frac{\langle k\rangle^2\langle k k_1\rangle_E}{\langle k^2\rangle^2} - 1, \end{equation*}$$
  where $\langle k^r\rangle$ is the r-moment of the degree and $\langle k k_1\rangle_E$ is the expected pairwise product of degrees over hyperedges.

Local measures. We consider the betweenness centrality and the community labels of nodes as local measures of higher-order structure.

• The node-based analogue of betweenness centrality for hypergraphs defined in [57] is
  $$\begin{equation*} BC\left(n\right) = \sum_{u \neq v \neq n\in V}\frac{\sigma_{uv}\left(n\right)}{\sigma_{uv}}, \end{equation*}$$
  where σ_uv is the number of shortest paths from node u to node v and $\sigma_{uv}(n)$ is the number of these shortest paths that pass through node n. Central nodes via this measure serve as 'bridge' nodes between different parts of a network.
• Community structure describes the mesoscale structure of complex systems by assigning the same labels to nodes that are densely connected with one another. To infer the community labels of nodes, we perform spectral clustering on the normalized Laplacian of the hypergraph, as proposed in [58]. We use a Hungarian matching algorithm to heuristically match two different sets of community labels.

Here, we analyze the email-enron dataset, a higher-order dataset generated from emails sent from a core set of employees at Enron [24, 59, 60], where nodes represent email addresses and edges represent email messages. Before analyzing this dataset, we removed isolated nodes, multi-edges, and singleton edges. We comment that our results are qualitatively similar to those computed when including these artifacts. The resulting dataset has 143 nodes and 1457 edges and has heterogeneous degree and edge size distributions.

We analyze this dataset using the four types of filtering that we introduced: the uniform, LEQ, GEQ, and exclusion filterings. This provides a broad overview of the different types of filtering that can be used to examine a dataset. The structural metrics that we consider are by no means exhaustive but provide an instructive example of how structure can change for different interaction sizes. To illustrate our filtering approach, we present the effective information, degree assortativity, and inferred community structure in this section. The interaction size affects not only the local structure of a higher-order network but the global structure as well. We start by analyzing the global structure presented in figure 2.

Zoom In Zoom Out Reset image size

As seen in figure 2(a), the trends in normalized effective information show substantial differences across the uniform, GEQ, LEQ, and exclusion filterings. While the normalized effective information largely decreases with k for the uniform, GEQ, and LEQ filterings, it largely increases for the exclusion filtering. However, these trends are not monotonic; for example, the normalized effective information for the GEQ filtering increases from k = 14 through $k = 16,$ while that for the exclusion filtering decreases from k = 16 to k = 17. Note that the effective information is not defined for $k = 14, 17, 18$ for the uniform filtering since no hyperedges of those sizes appear in the dataset.

In figure 2(b), we see that for both the uniform and GEQ filtering, the degree assortativity increases with interaction size before declining again. For small values of k, the assortativity fluctuates less for the GEQ filtering when compared with the uniform filtering because there are far more edges over which to average. In addition, the exclusion filtering illustrates that the pairwise interactions most significantly affect the degree assortativity. Lastly, the LEQ filtering demonstrates that we can capture most of the assortative structure with interactions of size ten and smaller.

In figure 3(a), we see that the betweenness centrality of all nodes is sensitive to the filtering parameter. Comparing the LEQ and GEQ filterings, we see that the centralities are much more stable when excluding higher-order interactions compared with excluding lower-order interactions. Furthermore, these centrality results provide fertile ground for data insights. For example, Bill Williams (node 137), an Enron trader, has the largest centrality in the uniform filtering for k = 2 but relatively small centralities for all other parameters. It turns out that he participated in many pairwise interactions (emails involving just one sender and recipient). Interestingly, his betweenness centrality remains high for a much larger parameter range for GEQ, LEQ, and exclusion filterings, indicating that he also serves as an intermediary for higher-order information. Another interesting trend is the very high betweenness centrality of Stanley Horton (node 46), President and CEO of Enron Transportation Services, in the GEQ filtering for intermediate values of k, indicating that he serves as a vital conduit of information in groups of intermediate size. In the other filterings, Stanley's centrality looks rather unremarkable across all filtering parameters.

Zoom In Zoom Out Reset image size

The community structure presented in figure 3(b) demonstrates that the assigned communities can change with differing interaction sizes. From the LEQ filtering, it seems that there are groups of nodes that switch memberships as larger interaction sizes are included. For the GEQ filtering, not only do we see nodes fail to join groups of a given size or larger, but their participation in a given community changes as well. In addition, the exclusion filtering indicates that interactions of a single size can be enough to change the community to which certain nodes belong. In contrast to existing literature suggesting that each node in a complex system has a single community label, relaxing this assumption leads to interesting trends in the community structure for different scales of interaction.

This case study illustrates the utility of filtering higher-order datasets by interaction size. Relaxing the assumption that all interactions contribute to a unified picture of a dataset can reveal structure that only exists at a particular scale. For all four measures of structure, we see noticeable changes when increasing the filtering parameter. Such changes can uncover the influence of different interaction sizes on the complete dataset.

In the appendices, we present multiple case studies across email, biological, and proximity domains. In the email and biological datasets, measuring effective information allows us to observe that higher-order interactions catalyze fewer unique associations than lower-order ones. Within the email datasets, taking interaction size as a proxy for gregariousness, we can identify key players at precise levels on the introvert/extrovert interaction scale. Stanley Horton stands as a compelling example from our Enron analysis, with his importance peaking within interactions of intermediate size. Further, among all the domains examined, community structure remains largely unchanged across all filterings in only the proximity datasets, with other domains showing complex changes in community structure with the filtering parameter.

We consider the effective information and degree assortativity as global measures of higher-order structure. Effective information was introduced in [26] for directed graphs and is defined as $EI = \mathcal{H}(\langle W^{out}_i\rangle) - \langle \mathcal{H}(W^{out}_i)\rangle$, where $W^{out}_i$ is the $\ell_1$-normalized vector of out-degrees for node i and $\mathcal{H}$ is the Shannon entropy. We extend this definition to hypergraphs by performing the degree computations on the clique projection associated with the weighted adjacency matrix $A = BB^T$ (setting the diagonal elements to zero), where B is the incidence matrix. The authors of [26] explain this metric as quantifying the strength of unique associations in a network.

Degree assortativity, the preferential tendency of nodes with similar degrees to connect with one another, has been extended to hypergraphs in several ways. In [15], the author defines the top-bottom, top-2, and uniform degree assortativity for non-uniform hypergraphs by selecting two nodes from a given hyperedge according to a given rule (the nodes with the smallest and largest degrees, the nodes with the two largest degrees, and two randomly selected nodes) and then computing the Pearson correlation coefficient. In our case, we approximate this metric by computing the assortativity for a sample of 10³ hyperedges.

In contrast to this method, [16] derives the dynamical assortativity for uniform hypergraphs with respect to a mean-field approximation of the largest eigenvalue as

$$\begin{equation*} \rho = \frac{\langle k\rangle^2\langle k k_1\rangle_E}{\langle k^2\rangle^2} - 1, \end{equation*}$$

where $\langle k^r\rangle$ is the r-moment of the degree and $\langle k k_1\rangle_E$ is the mean pairwise product of degrees over all possible 2-node combinations in each hyperedge in the hypergraph. We note that because [16] only defines dynamical assortativity for uniform hypergraphs, we can only compute it for the uniform filtering case (or whenever the filtering contains hyperedges of only one size, as is always the case for the largest parameter in GEQ).

The nodal structural properties that we consider are the centrality and community labels of nodes. There are many notions of centrality for hypergraphs [20, 21, 57]. In this study, we use a node-based analogue of the betweenness centrality for hypergraphs defined in [57] as

$$\begin{equation*} BC\left(n\right) = \sum_{u \neq v \neq n\in V}\frac{\sigma_{uv}\left(n\right)}{\sigma_{uv}}, \end{equation*}$$

where σ_uv is the number of shortest paths from node u to node v and $\sigma_{uv}(n)$ is the number of these shortest paths that pass through node n. Here, the number of shortest paths is computed on the unweighted clique projection. In all experiments, we report the $\ell_{\infty}$-normalized betweenness centrality to facilitate easy comparisons across filterings and filtering parameters.

There are many algorithms for community detection on hypergraphs [18, 19, 58, 62], and for this study, we use the method proposed in [58]. We start by defining the normalized Laplacian of the hypergraph as

$$\begin{equation*} L = \frac{1}{2}\left(I - D^{-1/2} A D^{-1/2}\right), \end{equation*}$$

where A is defined as before and $D = \text{diag}(A\textbf{1})$ is the diagonal matrix of degrees. We then compute the l largest eigenvectors of this matrix and perform k-means clustering on these vectors. One caveat is that the number of clusters is fixed to l, regardless of the number of connected nodes. This can sometimes lead to well-defined clusters being split into two or more clusters or other changes in cluster assignments.

To help visualize communities, we reorder nodes so that all nodes in the same community in the GEQ filtering for k = 2 appear together, and the largest communities appear on the bottom. To compare the communities output by the clustering algorithm across filtering parameters within a size-dependent filtering, we perform a weighted Jaccard-index-based Hungarian matching [63] to sequentially match current communities to previous communities as we sweep across the parameter. To compare communities across filterings, we reorder the nodes based on the ordering induced by the GEQ filtering and then match communities for k = 2. We then match across filtering parameters within the filtering in question as described above.

Consider a hypergraph with four nodes, two edges linking two disjoint pairs of nodes, and a hyperedge including all nodes. The exclusion filtering of the hypergraph with k = 2 and k = 4 is shown in figure 4. It is easy to verify that the hypergraph corresponding to k = 2 has a normalized effective information of $1 - \text{log}(3)/\text{log}(4),$ while that corresponding to k = 4 has a higher normalized EI of 1.

Zoom In Zoom Out Reset image size

For this metric, we use the sunflower hypergraph model as presented in [20] where we select the number of petals, n_p , the interaction size, k, and the number of nodes, c to be members of every petal. We populate each petal with the c center nodes and k − c isolated nodes. The resulting hypergraph has n_p edges and $(k - c)n_p + c$ nodes. We compute the terms in the expression for dynamical assortativity for each edge size k as follows:

$$\begin{align*} \langle d \rangle & = \frac{k n_p}{\left(k - c\right)n_p + c},\\ \langle d^2\rangle & = \frac{\left(k - c\right) n_p + c n_p^2}{\left(k - c\right)n_p + c},\\ \langle d d_1\rangle_E & = \frac{1}{\binom{k}{2}}\sum_{i < j} d_i d_j\\ & = \frac{1}{\binom{k}{2}}\left[ \binom{c}{2}n_p^2 + n_p\left(k - c\right)c + \binom{k-c}{2} \right], \end{align*}$$

and simplifying the final expression, we obtain

$$\begin{equation} \rho = -\frac{c\left(k-c\right)\left(n_p - 1\right)^2}{\left(k - 1\right)\left[k +c\left(n_p - 1\right)\right]^2}. \end{equation} \tag{ B1 }$$

From this expression, we can see that the hypergraph will always be disassortative because $k \gt c \unicode{x2A7E} 1$, and as the edge size increases, the low-degree nodes will 'dilute' the assortativity, eventually leading to a hypergraph with 0 assortativity. ee figure 5 for a visualization of this trend.

Zoom In Zoom Out Reset image size

Consider a star graph with eight edges joined together at a node with a sunflower having $n_p = 3$ and k = 3, as seen in figure 6. n the subhypergraph generated by the LEQ filtering for $k = 2,$ node 15 has an $\ell_\infty$-normalized betweenness centrality of 1 (and the rest 0). However, this drastically changes in the subhypergraph corresponding to k = 3, as the normalized centrality of node 15 decreases to 11/14, that of node 1 increases to 1, and that of node 2 increases to 24/35. Node 1 transitions from being the least to the most central node in the network.

Zoom In Zoom Out Reset image size

The hypergraph shown in figure 7 contains two dense communities (left, surrounded by dotted ovals), but when applying the GEQ filtering for k = 3 (right), two completely different communities emerge.

Zoom In Zoom Out Reset image size

Table 1 contains the summary statistics of all datasets considered in this study.

Table 1. Summary statistics of cleaned datasets.

Dataset	domain	$\lvert V \rvert$	$\lvert E \rvert$	Max. hyperedge size	Number of inferred communities
email-enron	email	143	1457	18	5
email-eu	email	986	24 520	40	42
disgenenet	biology	1816	7116	20	22
diseasome	biology	516	314	11	N /A
contact-primary-school	proximity	242	12 704	5	10
contact-high-school	proximity	327	7818	5	9

The 'email-eu' dataset is a higher-order dataset generated from email data sent within a large European research institution between October 2003 and May 2005 (18 months) [24, 59, 64, 65]. In this dataset, nodes represent email addresses and edges represent email messages. We neither account for the sender-receiver relationships nor the temporality of the emails sent. In addition, we remove duplicate emails involving the same group of people, emails sent to oneself, and email addresses that neither send nor receive emails. This dataset is heterogeneous in both the degree and edge size distributions.

In figure 8(a), we observe that for the effective information, small interactions contribute the most and that there is a sharp drop-off in effective information for interactions larger than 32 email recipients. We also observe in figure 8(b) that as the interaction size increases, the dataset becomes more and more strongly assortative. This is not the case for the top-bottom assortativity, but this could be caused by a highly dense core with a few peripheral nodes so that, on average, the degrees are highly correlated, but when looking at the highest and lowest degrees, they are highly negatively correlated. The community structure is harder to decipher; we have specified 42 communities to infer. Nonetheless, in figure 9(b), we see that as the email size increases, fewer and fewer nodes participate. We also see that excluding interactions of sizes two or three changes the community labels of many nodes. We see several communities persist over larger scales of interactions while other communities are much more sensitive to interaction size. In figure 9(a), we see that smaller interactions stabilize the centrality, and when excluding larger and larger interactions, the centrality is scale-dependent.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The 'disgenenet' dataset is a higher-order dataset describing the relationship between genes and the diseases associated with them [59, 66]. In this dataset, a disease is a node, and a gene is a hyperedge (the dual of the dataset available from XGI-DATA [59]). We have removed disease-disease correlations to enforce only disease-gene relationships. Before analyzing the dataset, we applied the LEQ filter with k = 20 for ease of presentation, as the maximum hyperedge size in the complete dataset is in the hundreds. We also excluded diseases with no associated genes and genes that have the same set of associated diseases.

In figure 10(a), the diseasome dataset shows decreasing effective information for increasing interaction size. Unlike the email-eu dataset, the pairwise interactions do not have as significant an influence, and excluding larger and larger interactions slightly increases the effective information. Figure 10(b) indicates that associations between diseases become assortative for those involving more than two diseases. The assortativity remains relatively stable, however, and plateaus for mid-sized interactions. As seen in figure 11(a), for all filterings, the most central node remains so across most interaction sizes, indicating that the diseases most associated with other diseases are consistent across all scales of interaction. Figure 11(b) shows nodes participating in many higher-order interactions, with many nodes not participating in lower-order interactions. While some communities persist across filtering parameters, many nodes often switch communities.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The 'diseasome' dataset tracks diseases and the genes associated with them [67]. In this dataset, a disease is a node, and a gene is a hyperedge. The 'label' attribute of the nodes is the disease description, and the 'label' attribute of the edges is the gene name. The disease-disease correlations were filtered out to enforce only disease-gene relationships. This results in a very sparse hypergraph, precluding us from analyzing community structure.

When ignoring interactions of smaller sizes through filtering, the effective information (shown in figure 12(a)) of the dataset decreases—although, interestingly, unlike the email datasets, the effective information drops much more sharply for the GEQ filtering, which may be due to the sparsity of larger interactions. This dataset exhibits low assortativity for many filterings and filtering parameters, as seen in figure 12(b). As in other datasets, the pairwise interactions seem to have the most significant effect on the assortativity. As seen from figure 13, centrality exhibits the most change in the smaller interactions, specifically those smaller than size 6.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The 'contact-primary-school' dataset is a contact network amongst children and teachers at a primary school in Lyon, France. This dataset was collected over two days and inferred via proximity sensing between wearable badges [24, 59, 68]. We form hyperedges through cliques of simultaneous contacts. Specifically, we construct a hyperedge for every maximal clique amongst the contact edges at each timestamp, and we aggregate all contacts to form a static higher-order network. We remove duplicate contact groups to sparsify the contact network because the dataset is extremely dense.

Proximity datasets can often have smaller ranges of interaction sizes not only because of physical limitations (in this study, proximity sensors detect interactions in the range of 1-1.5 meters [68]), but also because the method of reconstruction requires a complete clique to infer a hyperedge. We note that in figure 14(a), in contrast to the preceding datasets, the effective information is highest for larger interaction sizes. The assortativity shown in (figure 14(b)) seems to—on the whole—gradually increase with interaction si e. The most significant change is in the top-bottom assortativity, where interactions of size 4 are most disassortative. This indicates that the lowest and highest degrees are more anti-correlated, i.e. large-degree nodes tend not to interact with small-degree nodes. We see that the most central nodes are sensitive to pairwise interactions and that when excluding pairwise interactions, the centrality can change quite randomly with interaction size. We see that the community structure is relatively stable; if anything, figure 15(b) indicates that very few nodes participate in interactions larger than 4. This could be an artifact of the proximity data collection method.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

This is a temporal hypergraph dataset, which here means a sequence of timestamped hyperedges where each hyperedge is a set of nodes. This dataset is constructed from a contact network amongst high school students in Marseilles, France, in December 2013, inferred via wearable badges [24, 59, 69]. We form hyperedges through cliques of simultaneous contacts. Specifically, for every unique timestamp in the dataset, we construct a hyperedge for every maximal clique amongst the contact edges that exist for that timestamp. Timestamps were recorded in 20-second intervals, and we aggregated these contacts to form a static higher-order network. Because of the high density of this data, we remove duplicate interactions.

This dataset has similar limitations and artifacts due to the data collection method used for the contact-primary-school dataset. Similar to that dataset, interactions of size 4 have the most effective information as seen in figure 16, but in this case, interactions of size 5 have the least effective information. The assortativity, centrality, and community structure as seen in figures 16 and 17 yield similar conclusions to those of the contact-primary-school dataset.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size