Abstract
Hypergraphs offer an explicit formalism to describe multibody interactions in complex systems. To connect dynamics and function in systems with these higherorder interactions, network scientists have generalised randomwalk models to hypergraphs and studied the multibody effects on flowbased centrality measures. Mapping the largescale structure of those flows requires effective community detection methods applied to cogent network representations. For different hypergraph data and research questions, which combination of randomwalk model and network representation is best? We define unipartite, bipartite, and multilayer network representations of hypergraph flows and explore how they and the underlying randomwalk model change the number, size, depth, and overlap of identified multilevel communities. These results help researchers choose the appropriate modelling approach when mapping flows on hypergraphs.
Introduction
Researchers model and map flows on networks to identify important nodes and detect significant communities^{1,2,3,4,5,6}. From small to large system scales, random walkbased methods help to uncover the inner workings of the systems the networks represent^{7,8}. When standard network models with dyadic relations between pairs of nodes fail to adequately represent a system’s interactions, researchers turn to higherorder models of complex systems^{9,10}, including multilayer networks^{11,12,13,14} for multitype interactions, memory networks^{15,16,17} for multistep interactions and simplicial complexes^{18,19,20,21} and hypergraphs^{22,23,24,25} for multibody interactions.
While several methods can identify flowbased communities in multilayer^{11,26,27} and memory^{15,16,17} networks with higherorder Markov dynamics, researchers have focused on combinatorial methods to identify communities in hypergraphs^{28,29,30,31,32,33} and only recently begun to unravel flowbased community structures associated with random walks guided by hyperedges on hypergraphs^{25}. However, different systems and research questions call for different randomwalk and hypergraph models: random walks can be lazy, able to visit the same node multiple times in a row, or nonlazy and forced to move on. Hyperedges can have arbitrary weights, and nodes can have hyperedgedependent weights. Because these and other models can be represented with different network types—bipartite, unipartite and multilayer—the questions multiply: How do different hypergraph randomwalk models combined with different network representations change the flow dynamics at scales captured by communities?
For example, random walks on hypergraphs can model the flow of ideas in coauthorship networks. A node represents an author, and a hyperedge connects all authors of a paper. In the simplest dynamics, a random walker on a node picks a random hyperedge among those that contain the node and steps to a random node of the picked hyperedge. Then repeats. Excluding author selflinks for nonlazy walks or including hyperedge weights from paper citations or using hyperedgedependent node weights for varying author contributions are natural model variations that generate different dynamics^{23,24}. How does the organisation of authors in nested communities from research groups to research areas change with randomwalk model and representation? The many combinations of randomwalk models and representations available to address specific research problems require us to ask, for different data and different questions, which model and representation is best?
To address which combination of model and representation is best for answering different questions about various hypergraph data, we derive unipartite, bipartite and multilayer network representations of hypergraph flows with identical nodevisit rates for the same randomwalk model. For unique nodevisit rates when a representation requires directed links, we apply an unrecorded teleportation scheme robust to changes in the teleportation rate and that preserves the nodevisit rates when teleportation is superfluous in undirected networks^{34}. The informationtheoretic and flowbased community detection method Infomap^{35} allows us to explore how different hypergraph randomwalk models and network representation change the number, size, depth and overlap of identified multilevel communities. By analysing schematic and real hypergraphs, we find that the bipartite network representation requires the fewest links and enables the fastest community detection. A multilayer network representation that reinforces flows within similar layers gives the deepest modular structures with the most overlapping communities but at a high computational cost. The unipartite network representation provides a tradeoff between the two, with intermediate compactness, speed, and detectable modular regularities.
Results and discussion
Modelling flows on hypergraphs
We model flows on hypergraphs with random walks, using hypergraphs with nodes V, hyperedges E with weights ω, and hyperedgedependent node weights γ. Each hyperedge e has a weight ω(e). Each node u has a weight γ_{e}(u) for each hyperedge e incident to u, E(u) = {e ∈ E: u ∈ e}. To simplify the notation when normalising weights into probabilities, we denote node u’s total incident hyperedge weight d(u) = ∑_{e∈E(u)}ω(e) and hyperedge e’s total node weight δ(e) = ∑_{u∈e}γ_{e}(u)^{23}. With these weights, a lazy random walker moves from node u at time t to node v at time t + 1 in three stages by^{23}:

1.
Picking hyperedge e among node u’s hyperedges E(u) with probability \(\frac{\omega (e)}{d(u)}\).

2.
Picking one of the hyperedge e’s nodes v with probability \(\frac{{\gamma }_{e}(v)}{\delta (e)}\).

3.
Moving to node v.
Variations include nonlazy walks, which never visit the same node twice in a row with a modified second stage.

2.
Picking one of the hyperedge e’s nodes v ≠ u with probability \(\frac{{\gamma }_{e}(v)}{\delta (e){\gamma }_{e}(u)}\),
and teleporting walks, which jump to a random node at some rate to ensure that all nodes can be reached from any node in a finite number of moves, socalled ergodic walks. To model flows that tend to stay among similar hyperedges, such as among research papers with similar author lists and likely similar topics, we pick the next hyperedge based on its similarity to the previously picked hyperedge. These hyperedgesimilarity walks relate to link communities to reveal pervasively overlapping modules^{36} and neighbourhood flow coupling to reveal intermittent communities in temporal networks^{37}. Because hyperedgesimilarity walks depend on the previously picked hyperedge, they correspond to a higherorder Markov chain model.
These hyperedgesimilarity walks require multilayer networks since the other representations contain no information about the previously visited hyperedge^{26}. For example, compare the random walker in the unipartite and multilayer schematic networks in Fig. 1b, d: once the random walker reaches node c, only the multilayer network captures that the random walker came through the hyperedge with nodes c, f and g and can use different transition rates compared with arrival through the hyperedge with nodes a, b and c. Bipartite and unipartite networks, as well as multilayer networks, can represent the other randomwalk variations. Altering the randomwalk process alters the nodevisit rates, but a specific process has identical nodevisit rates irrespective of network representation by our design.
Bipartite networks offer the most direct representation of the basic threestage randomwalk process above. We represent the hyperedges with hyperedge nodes, and the three stages become a twostep walk between the nodes at the bottom and the hyperedge nodes at the top in Fig. 1b. For simplicity, we refer to them as nodes and hyperedge nodes. First a step from a node u to a hyperedge node e,
and then a step from the hyperedge node to a node v,
By starting the random walk on the nodes and taking two steps at a time, corresponding to a twostep Markov process^{38}, hyperedge nodes are only intermediate stops with zero flow when the random walk is back on the nodes after two steps. The stationary distribution of the random walk is concentrated to the nodes. For nonlazy walks represented with bipartite networks, we use socalled state nodes^{35} in the hyperedge nodes. We let each incoming link to a hyperedge node connect to a state node with outlinks to the hyperedge’s all nodes except the incoming link’s source node. This memory network ensures that walks are not backtracking^{39} (Fig. 2).
To represent the random walk on a unipartite network, we project the threestage randomwalk process down to a onestep process between the nodes and describe it with the transition rate matrix
where E(u, v) = {e ∈ E: u ∈ e, v ∈ e} is the set of hyperedges incident to both nodes u and v. Each hyperedge forms a fully connected group of nodes (Fig. 1c). Unipartite networks for nonlazy walks have no selflinks. The unipartite representation forms a weighted onemode projection of the bipartite representation and requires more links with its fully connected groups of nodes.
To represent the random walk on a multilayer network, we project the threestage randomwalk process down to a onestep process on state nodes in separate layers. Each hyperedge e with weight ω(e) forms a layer α with weight ω(α). A state node u^{α} represents u in each layer α ∈ E(u) that contains the node. All state nodes in the same layer form a fully connected set (Fig. 1d). The transition rate between state node u^{α} in layer α and state node v^{β} in layer β is
Node u’s state nodevisit rates in different layers sum to u’s visit rate in the unipartite and bipartite representations. With one state node per hyperedge layer that contains the node, the multilayer representation requires most nodes and links to describe the walk. But this cost from including state nodes such that all nodes have a state node for each incident hyperedge comes with benefits: the multilayer representation can describe higherorder Markov chains.
For example, to model flows that tend to stay among similar layers, we pick a hyperedge not only proportional to its weight but also proportional to its similarity to the hyperedge picked in the previous step. To include hyperedgedependent node weight information in the similarity measure, we use one minus the Jensen–Shannon divergence between the transition rate vectors P_{αv} and P_{βv} to nodes at layers α and β as the hyperedge coupling strength,
for β ∈ E(u, v). With node u’s total incident hyperedge weight in layer α
the hyperedgesimilarity walk has the transition rates
Because the transition rates at a node depend on the current layer, the random walks generate higherorder Markov dynamics that a unipartite or bipartite network representation without state nodes cannot capture.
To ensure ergodic nodevisit rates, we derived an unrecorded teleportation scheme that leaves the nodevisit rates unchanged when teleportation is superfluous for hypergraphs with hyperedgeindependent node weights, robust to changes in the teleportation rate when teleportation is needed^{34} and independent of the representation (see Methods).
Mapping flows on hypergraphs
To identify flowbased communities or modules in hypergraphs, we seek to compress a modular description of random walks on the network representations. We cast the problem of finding flowbased communities in hypergraphs as a minimumdescriptionlength problem with the map equation framework^{4}.
The map equation measures, in bits, the optimal codelength L per step of a random walk on a network for a given node partition M with m modules. When all nodes are in the same module, the map equation is simply the Shannon entropy H of the nodevisit rates \({\mathcal{P}}=\{{\pi }_{u}\}\). For the schematic example in Fig. 1 with lazy walks, the onemodule codelength is
for the bipartite, unipartite, and multilayer network representations because they have the same nodevisit rates. The modified hyperedgesimilarity walk gives slightly different nodevisit rates and codelength.
When the map equation combines within and betweenmodule codelengths in partitions with more than one module, different representations with identical nodevisit rates need no longer give the same codelength because the flows between modules can vary. For modules i = 1, …, m with
the map equation takes its general twolevel form
The first term is the codelength for betweenmodule movements, followed by the sum of codelengths for withinmodule movements over all modules.
When a network has modular regularities, a partition captures the modular flows when the random walker spends long times within the modules with few transitions between them. The codelength is shorter than in the onemodule solution because the information required to specify a random walker’s position in a module decreases with its size. But for partitions with too many modules, the information required for describing betweenmodule movements exceeds the gain from using small modules. The optimal partition has the shortest codelength. Its node assignment best captures the modular regularities of flows on the network.
Using the optimal threemodule solution for the unipartite network representation in Fig. 1c as an example, the codelengths for the bipartite representation—with the leftmost hyperedge assigned with nodes a, b and c in Fig. 1b to match the threemodule unipartite solution—and the unipartite representations are
with modules ordered from largest to smallest total flow rate. Since the nodevisit rates are the same, the higher betweenmodule flows for the bipartite representation
explain the large codelength difference. In the bipartite representation, a random walker can transition between modules even when visiting the same node multiple times in a row if an incident hyperedge belongs to a different module. Even with a zero nodevisit rate that does not contribute to the codelength, a hyperedge node with nodes in multiple modules costs extra bits because its links carry flows across module boundaries. As a result, the bipartite network representation favours fewer, larger modules than the unipartite network representation.
The multilayer representation enables further compression beyond the unipartite solution because a node’s state nodes can belong to different modules. The multilayer compression gain is illustrated for the nonlazy walk on the schematic hypergraph in Fig. 1. In this example, substituting nonlazy for lazy walks does not change the optimal unipartite solution, and the map equation takes the same form as in Eq. (11), but altered node and linkvisit rates change the codelength to 2.63 bits (Table 1). Assigning node f’s two state nodes f^{α} and f^{β} for its representation in the layers with nodes a, b, c and d, e, f, respectively, to modules two and three in the optimal multilayer solution changes Eq. (11) to
When modules two and three overlap in node f, less flow crosses their boundaries,
The compression gain from reduced flows between modules and within the third module is larger than the loss from adding state node f^{α} to the second module. Overlapping modules in the multilayer hyperedgesimilarity representation enable further compression because flows stay even longer within modules.
To find the optimal partitions for the different representations, we use the communitydetection algorithm Infomap^{35}. Infomap is to the map equation what the Louvain^{40} or the Leiden^{41} method is to the objective function modularity^{42}, which favours partitions with a high internal density of links compared with a statistical null model. Infomap uses a similar search algorithm as the Leiden method but tries to find the node assignment that minimises the map equation’s codelength. Infomap can find not only shallow twolevel partitions with nodes in modules, but also deeper hierarchical partitions—from toplevel supermodules with multiple levels of submodules down to leaflevel modules containing the nodes—if such multilevel solutions give higher modular compression^{43}. Infomap also finds twolevel or multilevel solutions in multilayer networks^{26}.
Using Infomap, we compare how much the different representations can compress modular flows. When mapping flows modelled by lazy and nonlazy random walks on the schematic network in Fig. 1, the optimal partitions of the bipartite networks have two communities. In contrast, the unipartite and multilayer networks have three communities and the multilayer networks with hyperedgesimilarity walks have four communities (Table 1 and Fig. 3).
With a state node for each hyperedge a node belongs to, the multilayer network provides Infomap with degrees of freedom that enables overlapping communities with possibly higher compression. But for this small network, only nonlazy walks give overlapping modules with 0.01 bits compression gain (Table 1). With walks that preferentially move to similar hyperedges, the optimal partitions of the multilayer hyperedgesimilarity network representations for lazy and nonlazy random walks both have more overlap in four modules (Table 1 and Fig. 3). The hyperedgesimilarity walks favour these overlapping modules because they stay longer within them than the regular walks.
For a given randomwalk model, the representations give equivalent nodevisit rates but alter the link flows, and with different link flows, the optimal partition can change. The bipartite network representation favours partitions with fewer modules than the unipartite network representation because assigning hyperedge nodes to modules implies encoding more transitions between modules. Multilayer representations, especially with walks that spend longer time among similar hyperedges, favour more overlapping modules. The randomwalk model determines how much the multilayer network modules overlap. Nonlazy and hyperedge similarity walks favour overlap because they lead to longer persistence times among nodes in possibly overlapping modules.
Experiments
To illustrate how the network representation affects detected communities in real hypergraphs, we generated a collaboration hypergraph from the 734 references in Networks beyond pairwise interactions: structure and dynamics by Battiston et al.^{10}. We modelled the referenced articles as hyperedges and their authors as nodes. Authors with multiple articles form connections between the hyperedges. We analysed the largest connected component with ∣V∣ = 361 author nodes in ∣E∣ = 220 hyperedges. The median number of authors in a hyperedge is 3, and the authors have contributed to 2.2 articles on average though most have only contributed to one.
Assuming that highly cited papers have higher influence and receive more flows^{23}, we assigned the relative importance of references by their number of citations c in December 2020. Some references had no citations and some were highly cited. One such example is Diffusion of innovations by Everett M. Rogers, with more than 120,000 citations. To avoid disproportionally large or small hyperedge weights ω(e), we weighted the edges by the logarithm of the number of citations and added unit constants to avoid the zero citation problem,
We modelled the authors’ different contributions to articles by assigning higher weights to the first and last author^{23}. We used the edgedependent node weights
We assumed equal contribution for alphabetically sorted authors, and assigned all of them weight γ(v) = 1. This model ranks a cocorresponding author’s contributions lower than those of the corresponding authors.
To study how hypergraph representations and randomwalk models affect the community structure, we generated bipartite, unipartite and multilayer representations for lazy and nonlazy random walks on the collaboration network. We identified nested hierarchical partitions in each network with Infomap, using 100 independent searches for each network. Infomap’s running time depends on the number of nodes, links and solution levels: the bipartite and unipartite representations finished 3–7 times faster than the multilayer representations. The nonlazy bipartite representation with many state nodes ran almost as long.
The optimised partitions for the lazy and nonlazy representations behave like the schematic example: The bipartite representations have the fewest leaf modules and highest codelengths, and the multilayer hyperedgesimilarity representations have the most leaf modules and shortest codelengths, with the unipartite and the regular multilayer representations in between (Table 2). Except for the nonlazy bipartite representation with its many state nodes, the lazy representations have more leaf modules and shorter code lengths than their corresponding nonlazy representations because the lazy random walk is more confined than the nonlazy random walk.
With more nodes than in the schematic example, the solutions have more depth. The bipartite solutions have three, and the unipartite and multilayer solutions have four hierarchical levels. The unipartite and multilayer solutions also have more top modules. With nonlazy dynamics, they split the largest top module, and in the lazy dynamics, they split the two largest top modules. But the secondlargest top module reunites in the hyperedgesimilarity representation, with stronger connections between similar hyperedges (Fig. 4 and Supplementary Fig. 1). The unipartite and multilayer solutions are also most similar at the leaf level (Supplementary Fig. 2).
In this larger example, the multilayer hyperedgesimilarity representations give more overlap. The nonlazy representations result in higher average overlap because random walkers visiting a node must continue to other nodes, often in the same or a similar hyperedge layer. When random walkers from dissimilar hyperedges come together at a node, they tend to return to where they came from and favour overlapping modules. The nonlazy representations also result in higher max overlap with the same authors topping all representations (Fig. 5).
In line with the informationtheoretic duality between finding regularities in data and compressing those data, representations that enable deeper solutions with more modules have shorter codelengths (Table 2). The lazy multilayer representation is an exception. Its optimised codelength is bound above by the lazy unipartite representation’s codelength—they have the same codelength for the same hard partition—and overlapping modules can potentially reduce the codelength. Infomap’s best codelength was instead 0.05% longer than for the lazy unipartite representation. Multilayer representations with their many state nodes and links aggravate the search problem, and Infomap could not find a better solution in 100 attempts. But the gain from overlapping modules is higher for the nonlazy multilayer representation and Infomap finds a solution with a significantly shorter codelength.
A case study on the fossil record
Palaeontologists classify major groups of marine animals archived in the fossil record into globalscale faunas that change over time^{44}. They have used standard^{45} and complex network representations^{46} to delineate these evolutionary faunas over the past 500 million years. However, it is still unclear how such an organisation of marine animals into modules representing largescale faunas changes with randomwalk model and input network representation.
To illustrate how the network representation of the underlying paleontological data affects empirical estimates of this macroevolutionary pattern, we generated a hypergraph from genuslevel fossil occurrences^{46} available from the Paleobiology Database^{47}. Due to computational limitations, we restricted our analysis to fossil occurrences from the Cambrian (541 MY) to the Cretaceous (66 MY). We modelled the remained 77 geological stages in the reduced data set as hyperedges and the 13,276 fossil genera as nodes. In this hypergraph, genera occurring in multiple geological stages form connections between hyperedges. We weighted the hyperedges by dividing the number of samples where a genus occurs in a given geological stage by the total number of samples recorded at the stage, a procedure modified from ref. ^{48}. We generated bipartite, unipartite and multilayer network representations for lazy and nonlazy random walks from the underlying palaeontology data and identified optimised partitions in the assembled networks with Infomap.
For lazy random walks, Infomap partitioned only the multilayer representations into multilevel communities, with three modules at the first hierarchical level reproducing the Cambrian, Paleozoic (with lowerlevel modules from Ordovician to Permian) and Mesozoic (with lowerlevel modules from Triassic to Cretaceous) largescale or evolutionary faunas^{44,46} (Fig. 6a). Like the schematic example and the hypergraph of metabolic reaction data, the bipartite representation for the lazy random walks has the fewest leaf modules and highest codelength. The multilayer hyperedgesimilarity representation has the most leaf modules, shortest codelength and highest overlap. Leaf modules in this representation can be interpreted as faunas from each geological period in the underlying data (Table 3).
For nonlazy random walks, Infomap partitioned the bipartite representation into a multilevel solution with shorter codelength than the unipartite representation and the standard multilevel representation (Fig. 6b). The multilayer hyperedgesimilarity representation also provides the most leaf modules and the highest overlap. Both multilayer representations reproduce the three largescale or evolutionary faunas. Unlike the other representations, the multilayer hyperedgesimilarity representation’s lowerlevel modules capture faunas from each geological period, including the Silurian.
Infomap applied to the bipartite representation of the nonlazy random walks identified similar lowerlevel faunas but combines Cambrian and Paleozoic into a single top module, obscuring the largescale pattern. For lazy and nonlazy random walk models, unipartite representations fail to capture the largerscale faunas that characterise the underlying system. Unipartite models also fail to distinguish some lowerlevel structures, providing a singlescale view of the system that lies between the lowest and higher levels in the multilayer solutions.
Our results suggest that representing fossil occurrence data with multilayer networks offers some advantages to quantify macroevolutionary patterns. Compared with unipartite and bipartite representations, multilayer networks enable discovering more regularities in the fossil record. Their optimised partitions provide higher compression, deeper hierarchy and a better multiscale view.
A case study on metabolic reaction data
Caenorhabditis elegans is an about 1mm long, transparent nematode found worldwide. C. elegans is one of the most studied model organisms in molecular biology for insights about diseases’ underlying metabolic pathways^{49,50,51}. We used the genomescale metabolic network model called iCEL1273^{52}, which contains 1273 genes, 623 enzymes and 1985 metabolic reactions and is available at wormflux.umassmed.edu. The data include metabolic pathways such as GlutRNA(Gln):Lglutamine amidoligase for AminoacyltRNA biosynthesis. The corresponding reversible reaction \({\rm{ATP}}\ +\ {\rm{GLN}}{\rm{L}}\ +\ {\rm{GLUTRNAGLN}}\ +\ {{\rm{H}}}_{2}{\rm{O}}\ \leftrightarrow \ {\rm{ADP}}\ + {\rm{GLNTRNA}}\ + {\rm{GLU}}{\rm{L}}\ +\ {\rm{H}}\ +\ {\rm{PI}}\)with reactants on the lefthand side and products on the righthand side requires one or more catalysing enzymes. The enzymes catalysing a reaction consist of proteins or protein complexes, which their coding genes’ Boolean logic can describe. For example, we denote the catalysing enzyme for the reaction above by C39B5.6 & Y66D12A.7 & Y41D4A.6, which corresponds to GlutamyltRNA(Gln) amidotransferase subunit B, GlutamyltRNA(Gln) amidotransferase subunit C and GlutamyltRNA(Gln) amidotransferase subunit A.
While standard networks with links between pairs of nodes representing reactants and products in the same reaction can provide insights about cell function^{3}, such dyadic relations fail to capture the coexistence of multiple proteins in complexes. Instead, we use hyperedges to represent metabolic reactions and nodes to represent reactants, products and enzymes. We represent each enzymatic protein complex with genes related by Boolean ANDs by a node such that genes related by Boolean ORs form multiple nodes in the same reaction. While many other abstractions of metabolic systems are possible, this representation naturally describes protein complexes in hypergraphs. To test how different randomwalk models and network representations capture functional modules of metabolites and enzymes, we generated unipartite, bipartite, and multilayer representations from the C. elegans hypergraph and identified multilevel communities with Infomap.
All hypergraph representations include modules with protein complexes otherwise overlooked in representations based on standard dyadic relationships. Again, the unipartite and multilayer representations have optimal solutions with shorter codelengths that reveal more modular regularities. The optimal solutions for the bipartite representations have fewer levels or modules (Table 4 and Fig. 7).
While the lazy and nonlazy random walk solutions are similar for several representations (Fig. 7a, b), the nonlazy walks give a deeper solution with more modules for the bipartite representation. Nevertheless, the solutions for the bipartite representations aggregate enzymes found in several metabolic processes, while the other representations include modules with enzymes representative of specific biological processes. For example, gene ontology enrichment analysis shows that Module 1:3 in the bipartite solution for nonlazy random walks includes both lipid and aminoacid metabolism. In the unipartite and multilayer representations, this module splits into distinct modules for lipid and aminoacid metabolism with more specific processes (Fig. 7b).
Only the multilayer hyperedgesimilarity solutions have significant overlap (Table 4). The module overlaps constitute common metabolites such as water and NAD. Assigning these common metabolites to multiple modules compresses the data more and reveals more regularities in smaller modules. But better representing the specific biological processes come at a relatively high computational cost. Infomap takes much longer to identify overlapping modules in the multilayer networks with numerous state nodes than hard partitions in the unipartite networks. Infomap even fails to compress the multilayer network beyond the unipartite network for nonlazy random walks because the more challenging search problem offsets the tiny compression gain from overlapping modules. The unipartite representation provides a good tradeoff between speed and compression, revealing more regularities than the bipartite representation much faster than the multilayer representations.
Conclusions
We have derived unipartite, bipartite, and multilayer network representations of hypergraph flows with different advantages. We used the informationtheoretic and flowbased community detection method Infomap to explore how different hypergraph randomwalk models and network representations change the number, size, depth and overlap of identified multilevel communities. By identifying flowbased communities both in a schematic and real hypergraphs—a small collaboration hypergraph of researchers working on networks beyond pairwise interactions, a large faunal hypergraph of sampled species across geological stages and the metabolic system of the model organism C. elegans—we found that the bipartite network representation enables the fastest community detection among the tested representations because it uses the fewest links and often has shallower solutions.
A multilayer network representation that reinforces flows within similar layers—one for each hyperedge—gave the deepest modular structures with the most module overlap. But the modular detection gain comes at a high computational cost: combining fully connected layers with other layers requires many more nodes and links than in the bipartite network representation. If the research question does not require hyperedge assignments or overlapping modules, the unipartite network representation provides a tradeoff with intermediate compactness, speed and the ability to reveal modular regularities. Among the randomwalk models, lazy walks typically give more modules in deeper nested structures, and nonlazy walks provide higher modular overlap. Our methods and results help researchers model and map flows on hypergraphs to study the effects of multibody interactions in complex systems.
Methods
Unrecorded teleportation
With hyperedgeindependent node weights where γ_{e}(u) = γ(u) for all hyperedges e ∈ E(u), undirected weighted networks can represent the dynamics, and the stationary distribution of the random walk π_{u} is proportional to the product of node u’s total incident hyperedge weight d(u) and weight γ(u). With normalised nodevisit rates^{23},
For the multilayer network representation, the nodevisit rates split between layers based on the node u’s incident hyperedge weight per layer state node
With hyperedgedependent node weights γ_{e}(u), only directed weighted networks can represent the dynamics. We use random teleportation to ensure ergodic walks when deriving the nodevisit rates with the poweriteration method. Unrecorded teleportation to links minimises the distortion^{34}: in each iteration of the poweriteration method, we distribute a fraction τ = 0.15 of each node’s flow volume among all nodes proportional to their outlink weights. The remaining flow volume moves on the links proportional to their weights. In the last iteration, we move all flows on the links proportional to their weights and record all flows on links and nodes to obtain the ergodic node and linkvisit rates with unrecorded teleportation. This procedure gives equivalent visit rates as simulating a random walker that only records moves on links: with probability 1 − τ, the random walker moves to a node by following the links proportional to their weights and records the link and the target node. With probability τ, the random walker teleports without recording to the link’s source node proportional to the link weight. The normalised number of recordings of each node and link gives the visit rates.
We want teleportation applied to undirected networks—where it is unnecessary—to leave the node and linkvisit rates unchanged. We achieve this smooth teleportation by scaling the transition rates from nodes by the nodevisit rates: then unrecorded teleportation proportional to the nodes’ total outlink weights followed by recorded moves on the links proportional to their weights distributes on the nodes according to the ergodic visit rates on undirected networks^{34}. For the general case when the node weights can depend on the hyperedge, and the network may be directed, we use Eq. (18) without assuming γ_{e}(u) = γ(u) as an approximation of the nodevisit rates:
for nodes and
for state nodes. With exact nodevisit rates, we would obtain the stationary flow volumes on links by multiplying the transition rates by the source nodes’ visit rates. With approximate nodevisit rates, instead, we obtain the link weights
for bipartite networks,
for unipartite networks, and
for multilayer networks. With unrecorded teleportation proportional to these link weights, modelling flows on hypergraphs give nodevisit rates pi_{u} and linkflow rates w_{uv} robust to changes in the teleportation rate and independent of the representation.
Module overlap metric
Modules overlap when Infomap assigns a node’s state nodes in the multilayer network representations to different modules. Measuring the overlap through the absolute number of assignments is misleading because the overlap is 2 regardless of the number of state nodes assigned to a different module than the rest. Instead, we used the effective number of assignments. If a fraction f of node u’s state nodes is assigned to the mth module in u’s module assignment set, the mth element of u’s assignment vector is \({a}_{m}^{u}=f\) and the effective number of assignments measured by the perplexity of u’s module assignments is
The effective number of assignments is one if all u’s state nodes are in one module, and it is equal to the number of assignments when the state nodes are divided evenly among u’s module assignments. We averaged over all nodes for the partition overlap.
Data availability
All data are available on GitHub (github.com/mapequation/mappinghypergraphs). The fossil data are available on the Paleobiology Database^{47} (paleobiodb.org). The metabolic reaction dataset for C. elegans, iCEL1273^{52}, is available at wormflux.umassmed.edu. Furthermore, all data are available from the corresponding author upon request.
Code availability
The source code is available on GitHub (http://github.com/mapequation/mappinghypergraphs).
Change history
28 June 2021
The original HTML version of this Article was updated shortly after publication to correct equations 12 and 15, which previously inadvertently included LaTeX \hrulefill commands.
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Acknowledgements
We thank Christopher Blöcker, Leyden Fernandez, Viktor Jonsson, Michael Schaub, Jelena Smiljanić and Alexander Vergara for valuable comments that helped us improve the manuscript. A. E was supported by the Swedish Foundation for Strategic Research, Grant No. SB160089. A. R., D. E. and M. R. were supported by the Swedish Research Council, Grant No. 201600796.
The computations were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at High Performance Computing Center North (HPC2N), partially funded by the Swedish Research Council through grant agreement no. 201805973.
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A.E. and M.R. conceived the study. A.E., A.R., D.E. and M.D. performed the numerical experiments and analysed the results. A.E. and M.R. wrote the manuscript.
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Eriksson, A., Edler, D., Rojas, A. et al. How choosing randomwalk model and network representation matters for flowbased community detection in hypergraphs. Commun Phys 4, 133 (2021). https://doi.org/10.1038/s4200502100634z
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DOI: https://doi.org/10.1038/s4200502100634z
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