To study shortest paths in temporal hypergraphs, we collected 24 publicly available data sets of real-world temporal systems with higher-order interactions. Specifically, our study focuses on hypergraphs that describe social interactions from a broad range of domains such as households and schools, hospitals, workspaces and conferences, as well as political interactions between individuals in Congress and even contact data from baboons.

We provide relevant summary statistics for the static versions of each data set in Table I, alongside the amount and resolution of successive time stamps for the temporal networks. A fuller description of the data sets is provided in the Methods section.

In simple graphs, relational information is described by nodes and edges, which encode pairwise interactions between pairs of nodes only. A path between 2 nodes and is an ordered, nonrepeating sequence of pairwise edges. Similarly, a hypergraph is a collection of nodes and hyperedges which encode interactions among an arbitrary number of nodes. Hyperedges are called pure pairwise or dyadic when they correspond to the classic notion of an edge by connecting only 2 nodes and are distinguished from truly higher-order interactions where 3 or more nodes participate in an interaction. The size of a hyperedge is defined as the number of nodes contained therein. A hyperpath or a higher-order path between 2 nodes in a hypergraph is defined as an ordered, nonrepeating sequence of hyperedges, where each subsequent step occurs between nodes within the same hyperedge.

The length of a higher-order path between any two nodes and , , is defined as the number of edges traversed. If no such path exists between 2 nodes, we set . Two nodes are connected if a path of finite length exists between them. While a pair of nodes can be connected by multiple distinct paths, the shortest path between any two nodes in a static network is defined as a path of minimum length, which might not necessarily be unique. Technical details regarding the exact computation of such hyperpaths are provided in the relevant section of Methods. Clearly, hyperpaths in higher-order networks will traverse hyperedges of varying sizes, where each segment of a hyperpath traverses a hyperedge of a potentially different size. As such, one may attach to each individual segment of a hyperpath its associated size , defined as the size of the hyperedge that is traversed during that particular step. In some cases, such a step may be contained in multiple hyperedges, in which case we choose the hyperedge of smallest size to define its size. Results concerning alternatives, such as selecting the mean or maximum in the case multiple hyperedges, and the frequency with which they occur are reported in Sec. 3 of the Supplemental Material [50]. Taken together, our analyses show that different strategies with regard to the hyperedge to traverse in case of redundancies do not induce a qualitative difference in our findings. Paths in higher-order networks contain rich information and one may calculate the average size of a path as the average size across traversed segments. For example, the average size of a path allows one to determine how much of a path makes use of dyads as opposed to genuinely higher-order ( ) interactions. The minimum size of a path is , which occurs when the hyperpath consists of purely pairwise interactions.

To illustrate these concepts, we consider the hypergraph in Fig. 1. A shortest hyperpath between nodes B and K has length . One of these is the hyperpath (B, F, H, K), where the segments (B, F), (F, H), and (H, K) have sizes 4, 5, and 2. The average size of such a hyperpath is then . We note that the segment ( ) is also contained in a hyperedge of size four but that we select the hyperedge of minimum size to assign a size. Importantly, other shortest hyperpaths of the same length can exist. For instance, here the hyperpath (B, D, H, K) also has the same . Another of such shortest hyperpaths is (B, D, G, K) which has a higher average size .

FIG. 1.

We are interested in studying the higher-order organization of shortest paths and the extent to which group interactions contribute to the forming of these paths in real-world systems.

We begin by considering static hypergraphs where we neglect information about the temporal nature of the interactions (see Methods for details). As an illustrative example, in Fig. 2, we show results for the social hypergraph from the well-known Copenhagen network study, which describes social interactions over 4 weeks between 576 university students on a university campus. Results of analyses on all other data sets are provided in the Supplemental Material, Secs. 2 and 3 [50].

FIG. 2.

In Fig. 2(a), we plot the distribution of shortest path lengths for higher-order paths in blue. We compare it with the distribution of path lengths of purely dyadic paths, obtained when taking considering only dyadic interactions, shown in gray. When higher-order interactions are not considered, we observe a distribution shift to the right, as such paths take longer to reach the same target as their higher-order counterparts. Interestingly, the same distributions of higher-order and dyadic path lengths are accurately reproduced by randomizing the data with a higher-order configuration model (see Methods section for details), as shown in Fig. 2(b).

Next, in Fig. 2(c), we study the average size of shortest paths between connected nodes in the higher-order network as a function of their length . We observe that shortest paths which are longer also tend to be those with a lower average size. This suggests that higher-order interactions are often central in the system, in agreement with recent work on hypercores [51] while dyadic edges remain crucial to connect more peripheral nodes. Due to the small sample (2) of maximum-length paths, the randomization does not reflect the downward trend at all lengths. To further corroborate our intuition, in Fig. 2(d) we plot the fraction of a shortest path that consists of purely dyadic segments, grouped by the length of the path itself. As expected, the curve is increasing, and such a behavior is again well reproduced in the randomized system. In Fig. 3 we extend the analysis of higher-order shortest paths to all our data sets. We observe that the average size as a function of decreases in all systems, except for a data set about social interactions in an office. We hypothesize that finite-size effects play a role for this workplace data, as the number of paths of length 3 are three orders of magnitude less than the other path length counts.

FIG. 3.

The inset of Fig. 3 shows for all data sets the difference in average path length for higher-order and purely dyadic paths. As expected, those systems which have in the main figure for all lengths are those which fall closest to the line in the inset. Finally, we observe that the lengths of shortest paths in all data sets are very short, never having a length of more than 5, in agreement with the small-world nature of complex networks [3].

In summary, while higher-order interactions are important to ensure connectivity in static hypergraphs, in such a simple scenario most shortest path features can be reproduced by preserving the higher-order degree distribution. As we will see in the next section, a much richer picture emerges once we move beyond static systems and expand our analysis to consider the role of time.

We model the temporal evolution of a networked system with a temporal network. A temporal network is a sequence of consecutive time-stamped static graphs, or alternatively a collection of nodes and a collection of time-stamped edges. At any time hyperlinks can be either active or inactive.

A temporal path is an ordered sequence of successive (edge, activation time) pairs, where consecutive pairs' time stamps must be increasing. An important consequence of temporality is that paths are not symmetric (even in the case of undirected graphs). In other words, the existence of a path from to does not imply the existence of a path from to and, in general, even when both exist, they might be different and have different lengths. In this work, we assume that a message cannot be propagated further than its direct neighbors within any single snapshot (although the alternative makes sense when time stamps are at longer intervals), and that a node can ‘‘retain’’ a message across multiple snapshots, following the approach of Ref. [13].

The concept of length in the temporal setting may be defined in two complementary ways. The temporal duration, denoted , of a temporal path from node to node is defined as the time elapsed between the time stamp at which node is first active and the time stamp at which node is active. By contrast, the length or step count, denoted , of a temporal path from nodes to node is defined as the total number of temporal edges traversed.

Shortest temporal paths can therefore be either those that have minimum duration ( ), or minimum length ( ). The fastest path (of shortest duration) between nodes and is the temporal path of minimum duration, and is set to infinity if such a path does not exist. The shortest path (with the fewest steps) counts the number of steps, and is hence analogous to the static case.

When a system describes time-varying interactions between groups of nodes, we make use of temporal hypergraphs. A temporal hypergraph is a sequence of static hypergraphs, indexed by time. The two notions of temporal length (duration and step count) can be defined as in the pairwise case. In Fig. 4, a simple temporal hypergraph illustrates how such measures can differ. A temporal path consisting of time-stamped links has a temporal length of 5 units, but a topological length of 4 links. In our temporal analysis, we mainly focus on fastest paths, although we present some results for both fastest and shortest in Sec. 4 of the Supplemental Material [50].

FIG. 4.

We investigate temporal connectivity and fastest paths in various real-world systems to discover how much of the system's connectivity is due to temporal higher-order interactions. Similarly to the static case, in Fig. 5, we begin our exploration of the temporal connectivity properties of real-world systems by focusing on the Copenhagen data set. As before, all analyses for the remaining data sets are available in Secs. 2 and 3 of the Supplemental Material [50], and we mention here only that similar trends are observable across the other data sets. In Fig. 5(a) we plot side by side the distribution of durations of fastest paths across all temporal interaction sizes in blue, and for only purely pairwise temporal interactions in gray. We note that the duration of such paths can be very long, even well above 100 time units. When only pairwise interactions are considered, there is a distinct distribution shift to the right of temporal path lengths. Such an effect of higher-order interactions is much greater than what we observed for the path length in static hypergraphs, making explicit the very relevant role of nondyadic connections in determining the connectivity properties of temporal systems.

FIG. 5.

In Fig. 5(b) we plot the same distribution of durations for a randomization of the data where hyperedges' time information is shuffled but the underlying static graph is kept constant. As shown, such a randomization fails to reproduce the distribution of durations of the actual system, as washing out temporal correlations among hyperedges makes the typical duration of fastest paths in the null model much shorter and narrower than in the real-world data.

Next, in Fig. 5(c) we plot the average size of fastest paths between connected nodes in the higher-order network, grouped by their duration, for both real and randomized data. As in the static case, we observe that paths that last longer also tend to have a lower average size in both the true and randomized networks. Faster paths traverse more higher-order temporal interactions. In contrast, paths with longer duration use more dyadic edges, as illustrated in Fig. 5(d), which shows the average fraction of fastest paths composed of pure dyadic temporal interactions, grouped by path duration. While the null model qualitatively reproduces the observed trends, we observe that fastest paths in the randomized data systematically underestimate the usage of dyadic edges in temporal paths.

In Fig. 6(a) we broaden our investigation to consider fastest paths for all data sets in our collection and plot against , the number of time steps across all data sets. Again, we observe again the same downward trend of decreasing average size with increasing path duration which is even more pronounced when plotting the topological length of fastest paths, and an increasing dyadic proportion for longer path lengths (last two not shown here). In Fig. 6(b), we calculate the proportion of node pairs in each data set that remain connected when dyadic edges are removed, i.e., when they are only allowed to traverse temporal interactions involving groups. While there is variability across real-world systems, we observe that in many cases a high percentage of paths disappear when only pure dyadic segments are considered, highlighting the importance of higher-order interactions for the global architecture of the system. For those nodes that remained connected even when only dyadic temporal interactions were allowed, in Fig. 6(c) we visualize how much longer on average a dyadic temporal path will tend to be than its higher-order counterpart. We note that in a some cases such differences can be very large, such as for the InVS13 and the Baboons data sets.

FIG. 6.

Path temporality also has consequences for the concept of connected components. Following Ref. [12], we define two nodes as strongly temporally connected if there is a temporal path from to and also vice versa. A temporal strongly connected component is the set of vertices for which any participating node can both reach and be reached by all other nodes in the set. To quantify the reachability of nodes in the systems, we study the temporal connectivity of the data sets in Table II by calculating the number of the strongly connected temporal components and the size of the largest. We compare this with the number and largest size of connected components in the static network. The temporal calculation is known to be an NP-complete problem [13] which is only computationally feasible for smaller graphs, so we present here results only for those networks that we could compute. We observe that the static hypergraph for nearly all data sets has a single giant connected component containing all nodes. In contrast, the temporal systems will be much more fractured, with components that are more numerous and smaller in size.

This section provides more detailed descriptions of the 24 data sets used in the above analyses.

Thirteen of the data sets [54–57] [Copenhagen, Friends & Family (monthly between August 2010–May 2011), Kenyan, Malawi] describe generic face-to-face interactions between individuals. The Friends & Family data set is particularly rich as it spans an extended time period while simultaneously storing information at a very high time resolution. We therefore opt to split it into 10 smaller data sets, each tracking the temporal evolution over one calendar month. Additionally, following the procedure for intermediate aggregation as described below, we group time frames into blocks of 8 h and create a static hypergraph for each interval.

Six of the data sets [58–60] (LyonSchool and Thiers13, HS11 and HS12, Elem1 and Mid1) describe interactions between pupils at school. Five data sets [58,61,62] describe interaction patterns within diverse work environments, namely, a hospital (LH10, DAWN), a workspace (InVS13 and InVS15), and a conference (SFHH). For all of these data sets, nodes correspond to individuals and hyperedges correspond to group interactions.

The final data set (Congress bills) describes political interactions between the U.S. House of Representatives and Senate [62]. Nodes here represent congresspersons and each hyperedge contains all sponsors and cosponsors of a specific legislative bill.

In Sec. 5 of the Supplemental Material [50], we additionally perform the same analyses on data describing coauthorship within geological journals across time, which is not inherently social but still a classic example of a setting in which higher order is natural.

We use 24 freely available data sets for our analyses describing group interactions. Even though interactions involve groups, the majority of the data sets store interactions between individuals as (time, node, node)-tuples as a computational simplification even when interactions involve groups. We are here interested in investigating the different contributions of higher-order and dyadic edges to the higher-order organization of networks. To this end, we assume that if individuals are all pairwise connected at time (i.e., they form a clique), then together they form a group of size and are promoted to a hyperedge of size in the temporal hypergraph.

Throughout our analyses, we distinguish between ‘‘pure’’ dyadic interactions and those which are an artifact of the data structures chosen to store the data digitally. We consider an edge to be a ‘‘pure pairwise’’ interaction when it does not form part of any larger clique, i.e., when it cannot be promoted to a group.

We create the static hypergraph by omitting temporal information and generating a hypergraph with all nodes and interactions reported. In effect, this collates all snapshots into a single network. This allows for potentially nested edges to exist in cases where a group of individuals at one time lose or gain members at an earlier or later time, even though in general we assume nested edges do not exist due to the way in which we create hyperedges.

To investigate the temporal evolution of social systems, we construct a temporal network consisting of a sequence of static hypergraphs, each associated with a specific time stamp. In some data sets we have very fine-grained temporal information with data logged at intervals of between 20 s and 5 min and which implies that many of the static hypergraphs are often extremely sparse. As a result, the temporal network will also be much more disconnected as nodes are active much less frequently. Following a procedure standard in the literature [10,63], we address this by performing a preprocessing step of coarse graining where we aggregate multiple smaller snapshots into a single bigger window (e.g., grouping together snapshots of resolution 30 s into groups of 5 min). The choice of aggregation window is determined in a data-driven manner by plotting the size of the largest connected component as a function of the size of the aggregation window and taking note of when its size saturates. We select the smallest window for which the largest connected component no longer grows.

The duration of a temporal path from node at some time to node is defined as the time elapsed between time and the first time that node appears in a snapshot (i.e., becomes active) after .

In the case that multiple shortest and fastest paths exist between a given node pair, we select a single path uniformly and at random. Whenever multiple hyperedges connect a single segment of a path, we select the hyperedge of minimum size and assign to the segment its size motivated by the intuition that smaller groups will more accurately transmit information as there are fewer places for information to flow towards [5]. We more fully study the impact of choosing the minimum instead of alternatives in Sec. 3 of the Supplemental Material [50].

Shortest paths in hypergraphs may be calculated by relating them to the graph shortest path problem via a clique expansion of the hypergraph.

Specifically, a shortest path between any node pair in a hypergraph is obtained by first transforming a static hypergraph to an undirected graph by placing a pairwise edge between 2 nodes whenever they share the same hyperedge (in effect, the clique expansion of the hypergraph). We then apply Dijkstra's shortest path algorithm [64] to find the shortest paths in the pairwise graph. Finally, we assign to each pairwise link in the shortest path the associated hyperedge that contains the pair.

Calculating fastest paths in temporal networks is a nontrivial task. To enable us to calculate time-respecting paths, we first map a temporal higher-order network to a static digraph representation whose nodes are now tuples of (time, node) pairs from the original temporal hypergraph [65,66]. We may then easily extract temporal paths from this new digraph. To calculate minimum-length topological paths, we assign a weight of 1 to all edges and employ Dijkstra's shortest path algorithm [64]. To calculate minimum-duration temporal paths, we weight edges by the time interval traversed and again use Dijkstra's algorithm. As before, in the presence of multiple paths we resolve ambiguities by picking one uniformly and at random. If a node ‘‘stores’’ a message across time steps of length , this node is considered to contribute units to the temporal path length.

Moreover, calculating shortest paths for large networks is computationally expensive, so we opt to pick a starting time and window length and label as the starting node set. In our case, all data sets have a window size of 300 time steps, except for the Friends & Family data set, which spans the entire calendar month. We then determine the existence and length of fastest and shortest paths from this starting set to all other nodes in the set within a timeframe of consecutive time snapshots. As before, nonexistent paths are set to have a path length of .

We consider randomizations of the data in both the static and temporal settings to compare our results. For both cases we generate 100 realizations of the particular null model and plot the resulting averages and 97.5% confidence intervals in the figures. For the static hypergraph, we use the higher-order configuration model, an extension of the well-known configuration model that keeps the degree distribution fixed across all orders and generates a new edge set. For temporal hypernetworks, we use a randomization that shuffles the time stamps of instantaneous events inside individual timelines to create a rearranged version while still preserving both the underlying static hypergraph and the total number of events, following the methodology of [67].