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Multiplex measures for higherorder networks
Applied Network Science volumeÂ 9, ArticleÂ number:Â 55 (2024)
Abstract
A wide variety of complex systems are characterized by interactions of different types involving varying numbers of units. Multiplex hypergraphs serve as a tool to describe such structures, capturing distinct types of higherorder interactions among a collection of units. In this work, we introduce a comprehensive set of measures to describe structural connectivity patterns in multiplex hypergraphs, considering scales from node and hyperedge levels to the systemâ€™s mesoscale. We validate our measures with three realworld datasets: scientific coauthorship in physics, movie collaborations, and high school interactions. This validation reveals new collaboration patterns, identifies trends within and across movie subfields, and provides insights into daily interaction dynamics. Our framework aims to offer a more nuanced characterization of realworld systems marked by both multiplex and higherorder interactions.
Introduction
From biological organisms to social groups, both natural and artificial systems demand sophisticated modeling tools to accurately capture their fundamental properties. Understanding how to represent interactions in such complex systems is crucial for unraveling their intricate architecture and emergent functionality. Networks have long offered a common language for studying these systems, representing units as nodes and interactions as dyadic links (Boccaletti etÂ al. 2006; Cimini 2019). However, this approach overlooks group interactions involving three or more nodes, which are essential in systems where higherorder interactions are prevalent (Battiston 2020, 2021; Torres etÂ al. 2021; Battiston and Petri 2022). Examples include collaboration networks (Patania etÂ al. 2017), human facetoface interactions (Cencetti etÂ al. 2021, folksonomies (Ghoshal etÂ al. 2009), species interactions within complex ecosystems (Grilli etÂ al. 2017), brain networks (Petri 2014), and cognitive associations (Citraro 2023).
Hypergraphs (Berge 1973), able to explicitly encode group interactions as hyperedges, have emerged as a popular framework to represent higherorder networks (Battiston 2020, 2021). Recently, significant effort has been dedicated to the characterization of hypergraphs, from centrality (Benson 2019; Tudisco and Higham 2021) and clustering (Benson etÂ al. 2018) measures, to backboning (Musciotto etÂ al. 2021, 2022) and reconstruction (Young etÂ al. 2021). Investigations into higherorder networks cover both micro (Lotito etÂ al. 2022, 2023; Lee etÂ al. 2020) and mesoscale levels (Wolf etÂ al. 2016; Vazquez 2009; Carletti etÂ al. 2021; Eriksson etÂ al. 2021; Chodrow etÂ al. 2021; Contisciani etÂ al. 2022; Chodrow etÂ al. 2023; Ruggeri etÂ al. 2023; Lotito etÂ al. 2024), unveiling structural principles essential to understanding group interactions in realworld systems. These higherorder interactions are known to impact the dynamic and collective phenomena within networked systems (Battiston 2021), affecting processes such as synchronization (Skardal and Arenas 2020; MillÃ¡n etÂ al. 2020; Lucas etÂ al. 2020; Gambuzza 2021; Zhang etÂ al. 2023), diffusion (Schaub etÂ al. 2020; Carletti etÂ al. 2020), spreading (Iacopini etÂ al. 2019; Chowdhary etÂ al. 2021; NeuhÃ¤user etÂ al. 2020) and evolution (AlvarezRodriguez 2021).
However, not all interactions in complex systems are alike; they may differ in nature, type, and scope. This observation led researchers to introduce the concept of multilayer and multiplex networks (Boccaletti 2014; KivelÃ¤ 2014), where links are encoded into different interaction layers, each representing a distinct type of relationship (De Domenico 2013; Battiston etÂ al. 2014). Multilayer and multiplex networks can successfully describe systems such as trade networks (Barigozzi etÂ al. 2010), transportations networks (Aleta etÂ al. 2017), collaboration networks (Battiston etÂ al. 2016), and the brain (De Domenico 2017). Multiplex hypergraphs, where layers encoding hyperedges of different type, could offer a robust tool for describing complex systems that involve group interactions of varying types. Despite a few exceptions (Vasilyeva 2021) and a significant potential, however, multiplex hypergraphs remain relatively unexplored, and a general set of tools for their analysis is still missing.
In this paper, we introduce some basic measures to characterize multiplex networks with higherorder interactions, spanning from the node/hyperedge level to the systemâ€™s mesoscale. We propose measures for the activity of nodes in different layers and orders, as well as for node degree correlation. Moreover, we partition nodes in generalists or specialists based on how their higherorder degree is spread across layers. We characterize hyperedges by examining hyperedge order distributions and overlap in multiple layers. Additionally, we introduce measures to quantify the layerdependent ability to connect either generalist or specialist nodes. Finally, we study the correlation of community and coreperiphery structures across layers.
We apply our proposals to three different realworld datasets: scientific coauthorship in physics, collaborations in movies and facetoface interactions in a high school. Our measures are able to highlight patterns of actorsâ€™ collaborations across various film genres, discern coauthorship trends within and across physics subfields, and offer insights into the daily interaction dynamics of students.
Multiplex hypergraphs
Multiplex hypergraphs model systems where interactions among units (i) may belong to multiple types and (ii) are not necessarily dyadic, i.e. they may involve more than two units. A multiplex hypergraph \({{\textbf {H}}}\) is defined as:
where each layer \(\alpha\) is a hypergraph \(H_{\alpha }(V, E_\alpha )\). Each hypergraph \(H_{\alpha }(V, E_\alpha )\) share the same set of entities V. \(E_\alpha \subseteq {\mathcal {P}}(V)\) is the set of all interactions of a specific type \(\alpha\). Moreover, we require \(e \ge 2\) for all \(e \in E_\alpha\) for any \(\alpha\). In other words, each layer in our framework shares the same set of nodes and represents a distinct set of interactions of the same nature. In Fig.Â 1, we show a simple multiplex hypergraph with 7 nodes and hyperedges spread across 3 layers.
We remark that our proposed framework is different from a multiplex representation of higherorder interactions where layers are defined by interactions of different order (Lucas etÂ al. 2020; Sun and Bianconi 2021).
In this work, we introduce a set of general tools to investigate multiplexity across different system scales in networks with higherorder interactions. We validate our measures and discuss relevant findings for three distinct realworld datasets:

aps (Coauthorship network), where nodes are authors, and hyperedges represent groups of authors who have coauthored a paper. Each layer collects papers from the same subfield of physics, identified by a PACS code (Physics and Astronomy Classification Scheme) (Dataset 2021).

imdb (Costarring network), where nodes represent actors, and hyperedges represent the cast of a specific movie. Each layer corresponds to a movie genre.

high school (Social network), where nodes are students, and hyperedges represent groups of students interacting facetoface, with each layer grouping interactions from the same day of the week (Mastrandrea etÂ al. 2015).
Detailed statistics about datasets are provided in TableÂ 1.
Node properties
We begin by investigating multiplex properties at the node level. The first basic measure we consider is node activity (Nicosia and Latora 2015). A node i is active at layer \(\alpha\) if i participates in at least one interaction at layer \(\alpha\). FigureÂ 2 shows statistics on nodesâ€™ simultaneous activity across multiple layers. Specifically, the yaxis plots the proportion of nodes (from the total node count) active in at least x layers. By definition, these curves exhibit a decreasing trend, with variations in the negative slopes reflecting the datasetsâ€™ diversity. While it is uncommon for scientists and actors to be active across more than 2 or 3 layers, students tend to be active in all layers. In fact, the inactivity of a student in a specific layer implies their absence from school on that day.
So far, we have grouped all interactions together, regardless of their order. To obtain more detailed insights about higherorder interactions, we can examine node activity for each specific interaction order d. To this scope, we introduce a list \({{\textbf {A}}}\) of node activity matrices, one for each node i:
Similarly, one can define activity from a layer perspective and consider a list \({{\textbf {B}}}\) of layer activity matrices, one for each layer \(\alpha\):
It can be useful to aggregate information about nodes and define an aggregated layer activity matrix L as:
In Fig.Â 3a, each dataset is represented as a graph where vertices are the layers of the multiplex hypergraph and links measure the similarity in activity patterns of two layers \(\alpha\) and \(\beta\), quantified as the cosine similarity of their node activity matrices \({{\textbf {B}}}_\alpha\) and \({{\textbf {B}}}_\beta\). The thicker the link, the higher the similarity. This figure emphasizes layers that not only share common active nodes, but also exhibit similar patterns of participation across different hyperedge orders. Particularly, a consistent higherorder similarity is observed across school days, reflecting recurring interaction patterns throughout the week. Other datasets show more heterogeneous behaviour, with documentary casts differing significantly from other layers, while drama and comedy casts exhibit similar patterns.
FigureÂ 3b shows the aggregated layer activity matrices L for the three datasets. To account for variations in layer size, we normalize each row by the total number of active nodes in the respective layer. Distinct collaboration patterns emerge across the subfields of physics and movie genres. For instance, scientists in General or Electromagnetic Physics usually contribute to papers with a smaller number of coauthors, whereas coauthor groups in fields like Elementary Particles and Nuclear Physics exhibit more variation in size. In movie collaborations, actor activity is concentrated in mediumsize groups, typically between ten and twenty members. However, documentaries often feature smaller casts, while family and comedy movies tend to have larger ones. The figure once again highlights how students at school maintain a consistent group size in their interactions throughout the week.
Similar to node activity, node degree (defined as the number of interactions in which a node participates) is another property that can be used to measure the activity across the different layers and interaction orders. We define a list \({{\textbf {K}}}\) of node degree matrices, one for each node i:
We use \(k_{i \alpha }\) to denote the total number of interactions involving i in layer \(\alpha\), irrespective of their order:
where D is the order of the largest interaction in the dataset.
In Fig.Â 4a, we analyze the correlation of node degree across layers, exploring the extent to which a node with a high or low degree in one layer similarly exhibits a high or low degree in another layer. The correlation matrix for physics collaborations uncovers a hierarchical structure, with strong correlations among specific subfields sharing commonalities and notable scientists, such as in Nuclear and Elementary Particles Physics. In contrast, the degree correlations among actors are generally weak, though certain genres, like thriller and horror, show similarities. A significant correlation in node degrees across consecutive days in high school suggests stable and structured daily interaction patterns, implying that individuals with numerous interactions on one day tend to maintain similar levels of interactions on subsequent days, and vice versa.
We now define the overlapping degree \(o_i\) for a node i as the total number of interactions involving i, irrespective of both layers or orders:
It can be interesting to measure (i) how the overlapping degree of a node i is spread across the layers, i.e., if the degree is concentrated in certain layers or if it is uniformly distributed; (ii) how interactions involving node i are spread across orders. We measure (i) by defining the participation coefficient \(P_i\) of a node i of the degree with respect to the layers:
where \(k_{i\alpha }\) is the degree of node i at layer \(\alpha\), \(o_i\) is the overlapping degree of node i and M is the total number of layers. We measure (ii) by considering the average order of the interactions node i participates in.
In Fig.Â 4b, we represent each unit i of the different systems on a Cartesian plane, characterizing them across three distinct dimensions: their overlapping degree \(o_i\) (on the yaxis), their participation coefficient \(P_i\) (on the xaxis), and the average order of the interactions in which they are involved (indicated by color intensity). In general, such three dimensions provide different information about connectivity patterns and are only weakly correlated, or even uncorrelated. In aps, scientists are spread across the plane in terms of degrees and average interaction order, displaying an overall tendency towards specialization in a selected number of physics subfields, yet the behavior remains heterogeneous. imdb displays isolated outliers with a very high degree, low dispersion around the average interaction order, and a general tendency towards uniform participation across multiple genres. high school shows students covering the entire spectrum of node degrees and average group orders, with an expected tendency to interact uniformly across school days.
Hyperedge properties
We now turn our attention to the properties of the hyperedges encoding interactions in the same three realworld systems.
We begin by considering the simplest measure for characterizing higherorder interactions, namely, the order of the groups. In Fig.Â 5, we plot the hyperedge order distribution disaggregated by layers. aps and imdb reveal heterogeneity across layers, suggesting that different physics subfields and movie genres exhibit distinct patterns of collaboration in terms of the number of people involved in a paper or a movie cast. For example, genres such as documentaries and animated movies typically feature fewer actors compared to other genres. Conversely, papers in Elementary Particles and Nuclear Physics often include a larger number of authors compared to those in other areas of physics. The distributions in high school are stable across layers, indicating that patterns of facetoface interactions tend not to change over the days, with a general preference for smaller groups over larger ones.
Another property frequently studied in the context of multiplex networks is edge overlap, which measures the extent to which interactions among the same nodes tend to repeat across multiple layers. We define hyperedge overlap as the maximum number of layers in which an interaction repeats exactly. In Fig.Â 6, we present the distribution of hyperedge overlap, including information about the order of the interactions. As expected, aps displays a high degree of hyperedge overlap, indicating that the same set of scientific authors consistently interact across multiple areas of physics. Conversely, for actors, hyperedge overlap decays very rapidly. Small interactions typically exhibit a higher degree of overlap than large interactions. Historically, edge overlap in multiplex networks with higherorder interactions has often been investigated by projecting hyperedges at different layers into cliques, frequently resulting in extremely high values of edge overlap. Our analysis suggests that patterns of hyperedge overlap are more complex and that projections of hyperedges can account for the high amount of overlap previously observed (Battiston etÂ al. 2014).
Finally, we assign a score \(P_e\) to each hyperedge e, defined in terms of the participation coefficient of the nodes involved in the interaction:
where e represents the number of nodes participating in hyperedge e and \(P_{i}\) is the participation coefficient of node i, as defined in the previous section.
This measure captures the tendency of hyperedges to connect nodes that either specialize in a few layers or act as generalists across multiple layers. In Fig.Â 7, we use boxplots to show distributions of P for hyperedges in various layers. It is noteworthy that layers can display heterogeneous mean values for the participation coefficient of their hyperedges. For example, casts in documentaries and coauthors in Nuclear and Elementary Particles Physics tend to include specialists. On the other hand, family and thriller movies are more likely to feature generalist actors. In high school, layers exhibit a consistent maximum mean value for the participation coefficient of hyperedges, attributed to studentsâ€™ regular attendance at school each day.
Mesoscale properties
We finally shift our focus towards mesoscale structures, examining the emergence of communities and coreperiphery structures within different layers of realworld hypergraphs.
Communities are groups of nodes that display a higher degree of connectivity among themselves than with the rest of the nodes in the system. In hypergraphs, a community is defined as a subset of nodes that tend to form cohesive units by participating in common hyperedges. When analyzing multiplex systems, it is typical to examine the similarities in community structures observed across various layers. In this direction, we employ a method for hard clustering, applied independently to each layer, which is an extension of the wellestablished Infomap algorithm to the case of hypergraphs (Eriksson etÂ al. 2021). In general, Infomap minimizes the map equation, which quantifies the description length required to represent the random walkerâ€™s movements on the network (Rosvall etÂ al. 2009). This optimization effectively partitions the network into communities that best capture the inherent modular structure. The method is publicly available (Eriksson etÂ al. 2021) and we have used default parameters.
To assess the similarity and consistency of community structure across different hypergraph layers, we use Normalized Mutual Information (NMI), taking into account the set of nodes active in both layers. NMI is defined as:
where \(I(C_1, C_2)\) denotes the mutual information between partitions \(C_1\) and \(C_2\). Mutual information quantifies the amount of information shared between the two partitions, i.e., measures how much knowing the community structure in one partition informs about the structure in the other. \(H(C_1)\) and \(H(C_2)\) represent the entropies of the partitions \(C_1\) and \(C_2\), respectively. By normalizing the mutual information \(I(C_1, C_2)\) with the geometric mean of these entropies, NMI adjusts for the variability in partition sizes and the number of communities, allowing for a fair comparison of community structures across different partitions. NMI ranges from \(0\) (indicating no mutual information) to \(1\) (indicating perfect agreement).
In Fig.Â 8a, we present the outcomes of this analysis through heatmaps, highlighting the strength of community structure similarities across layers. For example, community structures in animation and fantasy movie casts are closely related, as are those in comedy and family movie genres, whereas documentary casts show a completely uncorrelated structure. The community structure within high school interactions remains consistent across days. Physics collaborations reveal a significant degree of similarity across fields, though some layers exhibit more pronounced similarities than others. Overall, communities tend to be preserved within physics subfields and school days, while movie genres often demonstrate predominantly uncorrelated communities.
We then direct our attention to coreperiphery structures. Coreperiphery structures delineate the existence of a group of central and tightly connected nodes, the core, surrounded by less densely connected peripheral nodes, forming a distinctive organizational pattern often crucial for system functionality (Borgatti and Everett 2000).
We detect coreperiphery structure for each layer independently using a method tailored for hypergraphs (Tudisco and Higham 2023). Such a method ranks nodes assigning to each node i a value \(c_i\), where \(c_i\) is a real number within the range of 0 to 1. This value delineates the extent to which a node participates in the core (value closer to 1) or the periphery (value closer to 0) structure of the system. Vector \({\textbf{c}}\) is selected for each layer \(\alpha\) independently. Following the work by Tudisco and Higham (Tudisco and Higham 2023), for each layer \(\alpha\) we select the vector \({\textbf{c}}\) that optimizes the following function:
This continuous scale allows for a nuanced characterization of each nodeâ€™s role within the coreperiphery framework. Additionally, by comparing the coreness score \(c_i\) for each node \(i\) across the different layers of a multiplex hypergraph, we can analyze the variation or consistency of a node centrality across the layers. This method is publicly available (Tudisco and Higham 2023). We use the implementation from Hypergraphx (Lotito 2023).
To provide a visually appealing way of highlighting correlations of coreperiphery structures and node behaviour across layers, Fig.Â 8b shows heatmaps in which rows are layers and columns are nodes, and each entry is coloured depending on node coreness. To compare the coreness value of single nodes across layers and visualize to which extent it keeps its core value, we maintain a consistent sorting of the nodes on the xaxis. For each dataset, nodes are sorted according to their core values in the aggregated hypergraphs (i.e., the hypergraph obtained by dropping information about layers and collapsing every hyperedge to a single layer). We observe that coreness values of nodes are maintained across layers exhibiting patterns similar to those seen in community structure correlations.
Conclusions
Networks have established themselves as a fundamental tool in a variety of disciplines to encode and study systems of interacting units. With the idea of capturing richer information about interactions, novel and more comprehensive network models have emerged: (i) multiplex networks, describing links of different types, and (ii) hypergraphs, encoding nondyadic ties. Bridging these two notions, in this work we have introduced a general set of measures to characterize the structure of multiplex hypergraphs at multiple scales. We introduced a description of nodes in terms of higherorder activity patterns and degrees, to quantify the extent and magnitude of node participation in interactions of different orders across layers. Nodes have been also characterized in terms of how their degree is correlated and spread in the different layers, and by their preferred order of group interactions. For hyperedges, we have studied their order distribution disaggregated by layers, highlighting different patterns of group interactions depending on the hyperedge type. We have quantified the extent to which hyperedge tends to repeat exactly in multiple layers and we have analyzed the layerdependant property of hyperedges of linking nodes with low or high participation coefficients. Finally, we have analyzed hypergraphs at their mesoscale, quantifying similarities of communities and coreperiphery participation across layers. We have validated our proposed measures on three datasets from different domains, describing collaboration patterns across physics subfields, movie genres and daily interactions among students.
In summary, we believe that our measures can be useful in describing the structure of various empirical datasets characterized by both multiplex and higherorder interactions. We also hope that this initial characterization of multiplex hypergraphs will spark interest from a methodological perspective, such as proposing frameworks for extracting multiplex communities in hypergraphs. Further characterizations could be enhanced by considering the complex patterns of temporality in hyperedges, a common feature of higherorder systems (Cencetti etÂ al. 2021; Ceria and Wang 2023; Gallo etÂ al. 2023; Iacopini etÂ al. 2023; Di Gaetano etÂ al. 2024; ArreguiGarcÃa etÂ al. 2024; Mancastroppa etÂ al. 2024; Chodrow 2020).
Data Availability
Data is publicly available as part of HGX data repository (Lotito 2023).
Code Availability
Multiplex measures for higherorder networks, as well as basic functions for handling multiplex hypergraphs, are available as part of Hypergraphx (HGX) (Lotito 2023).
References
Aleta A, Meloni S, Moreno Y (2017) A multilayer perspective for the analysis of urban transportation systems. Sci Rep 7:44359
AlvarezRodriguez U et al (2021) Evolutionary dynamics of higherorder interactions in social networks. Nat Hum Behav 5:586â€“595
ArreguiGarcÃa B, Longa A, Lotito QF, Meloni S, Cencetti G (2024) Patterns in temporal networks with higherorder egocentric structures. arXiv:2402.03866
Barigozzi M, Fagiolo G, Garlaschelli D (2010) Multinetwork of international trade: a commodityspecific analysis. Phys Rev EStat Nonlinear Soft Matter Phys 81:046104
Battiston F et al (2020) Networks beyond pairwise interactions: structure and dynamics. Phys Rep 874:1â€“92
Battiston F et al (2021) The physics of higherorder interactions in complex systems. Nat Phys 17:1093â€“1098
Battiston F, Iacovacci J, Nicosia V, Bianconi G, Latora V (2016) Emergence of multiplex communities in collaboration networks. PLoS ONE 11:e0147451
Battiston F, Nicosia V, Latora V (2014) Structural measures for multiplex networks. Phys Rev E 89:032804
Battiston F, Petri G (2022) Higherorder systems. Springer
Benson AR (2019) Three hypergraph eigenvector centralities. SIAM J Math Data Sci 1:293â€“312
Benson AR, Abebe R, Schaub MT, Jadbabaie A, Kleinberg J (2018) Simplicial closure and higherorder link prediction. Proc Natl Acad Sci 115:E11221â€“E11230
Berge C (1973) Graphs and hypergraphs. NorthHolland Pub. Co
Boccaletti S et al (2014) The structure and dynamics of multilayer networks. Phys Rep 544:1â€“122
Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang DU (2006) Complex networks: structure and dynamics. Phys Rep 424:175â€“308
Borgatti SP, Everett MG (2000) Models of core/periphery structures. Soc Netw 21:375â€“395
Carletti T, Battiston F, Cencetti G, Fanelli D (2020) Random walks on hypergraphs. Phys Rev E 101:022308
Carletti T, Fanelli D, Lambiotte R (2021) Random walks and community detection in hypergraphs. J Phys Complex 2:015011
Cencetti G, Battiston F, Lepri B, Karsai M (2021) Temporal properties of higherorder interactions in social networks. Sci Rep 11:1â€“10
Ceria A, Wang H (2023) Temporaltopological properties of higherorder evolving networks. Sci Rep 13:5885
Chodrow PS (2020) Configuration models of random hypergraphs. J Complex Netw 8:cnaa018
Chodrow P, Eikmeier N, Haddock J (2023) Nonbacktracking spectral clustering of nonuniform hypergraphs. SIAM J Math Data Sci 5:251â€“279
Chodrow PS, Veldt N, Benson AR (2021) Generative hypergraph clustering: from blockmodels to modularity. Sci Adv 7:eabh1303
Chowdhary S, Kumar A, Cencetti G, Iacopini I, Battiston F (2021) Simplicial contagion in temporal higherorder networks. J Phys Complex 2:035019
Cimini G et al (2019) The statistical physics of realworld networks. Nat Rev Phys 1:58â€“71
Citraro S et al (2023) Hypergraph models of the mental lexicon capture greater information than pairwise networks for predicting language learning. New Ideas Psychol 71:101034
Contisciani M, Battiston F, De Bacco C (2022) Inference of hyperedges and overlapping communities in hypergraphs. Nat Commun 13:1â€“10
De Domenico M et al (2013) Mathematical formulation of multilayer networks. Phys Rev X 3:041022
De Domenico M (2017) Multilayer modeling and analysis of human brain networks. Giga Sci 6:gix004
Eriksson A, Edler D, Rojas A, de Domenico M, Rosvall M (2021) How choosing randomwalk model and network representation matters for flowbased community detection in hypergraphs. Commun Phys 4:1â€“12
Di Gaetano L, Battiston F, Starnini M (2024) Percolation and topological properties of temporal higherorder networks. Phys Rev Lett 132:037401
Gallo L, Lacasa L, Latora V, Battiston F (2023) Higherorder correlations reveal complex memory in temporal hypergraphs. arXiv:2303.09316
Gambuzza LV et al (2021) Stability of synchronization in simplicial complexes. Nat Commun 12:1â€“13
Ghoshal G, ZlatiÄ‡ V, Caldarelli G (2009) Random hypergraphs and their applications. Phys Rev E 79:066118
Grilli J, BarabÃ¡s G, MichalskaSmith MJ, Allesina S (2017) Higherorder interactions stabilize dynamics in competitive network models. Nature 548:210â€“213
Iacopini I, Petri G, Barrat A, Latora V (2019) Simplicial models of social contagion. Nat Commun 10:1â€“9
Iacopini I, Karsai M, Barrat A (2023) The temporal dynamics of group interactions in higherorder social networks. arXiv:2306.09967
KivelÃ¤ M et al (2014) Multilayer networks. J Complex Netw 2:203â€“271
Lee G, Ko J, Shin K (2020) Hypergraph motifs: concepts, algorithms, and discoveries. Proc VLDB Endow 13:2256â€“2269
Lotito QF et al (2023) Hypergraphx: a library for higherorder network analysis. J Complex Netw 11:cnad019
Lotito QF, Musciotto F, Montresor A, Battiston F (2022) Higherorder motif analysis in hypergraphs. Commun Phys 5:79
Lotito QF, Musciotto F, Montresor A, Battiston F (2024) Hyperlink communities in higherorder networks. J Complex Netw 12:cnae013
Lotito QF, Musciotto F, Battiston F, Montresor A (2023) Exact and sampling methods for mining higherorder motifs in large hypergraphs. Computing, pp 1â€“20
Lucas M, Cencetti G, Battiston F (2020) Multiorder Laplacian for synchronization in higherorder networks. Phys Rev Res 2:033410
Mancastroppa M, Iacopini I, Petri G, Barrat A (2024) The structural evolution of temporal hypergraphs through the lens of hypercores. arXiv:2402.06485
Mastrandrea R, Fournet J, Barrat A (2015) Contact patterns in a high school: a comparison between data collected using wearable sensors, contact diaries and friendship surveys. PLoS ONE 10:e0136497
MillÃ¡n AP, Torres JJ, Bianconi G (2020) Explosive higherorder Kuramoto dynamics on simplicial complexes. Phys Rev Lett 124:218301
Musciotto F, Battiston F, Mantegna RN (2021) Detecting informative higherorder interactions in statistically validated hypergraphs. Commun Phys 4:1â€“9
Musciotto F, Battiston F, Mantegna RN (2022) Identifying maximal sets of significantly interacting nodes in higherorder networks. arXiv:2209.12712
NeuhÃ¤user L, Mellor A, Lambiotte R (2020) Multibody interactions and nonlinear consensus dynamics on networked systems. Phys Rev E 101:032310
Nicosia V, Latora V (2015) Measuring and modeling correlations in multiplex networks. Phys Rev E 92:032805
Patania A, Petri G, Vaccarino F (2017) The shape of collaborations. EPJ Data Sci 6:1â€“16
Petri G et al (2014) Homological scaffolds of brain functional networks. J R Soc Interface 11:20140873
Rosvall M, Axelsson D, Bergstrom CT (2009) The map equation. Eur Phys J Special Top 178:13â€“23
Ruggeri N, Contisciani M, Battiston F, De Bacco C (2023) Community detection in large hypergraphs. Sci Adv 9:eadg9159
Schaub MT, Benson AR, Horn P, Lippner G, Jadbabaie A (2020) Random walks on simplicial complexes and the normalized hodge 1laplacian. SIAM Rev 62:353â€“391
Skardal PS, Arenas A (2020) Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching. Commun Phys 3:1â€“6
Sun H, Bianconi G (2021) Higherorder percolation processes on multiplex hypergraphs. Phys Rev E 104:034306
Torres L, Blevins AS, Bassett D, EliassiRad T (2021) The why, how, and when of representations for complex systems. SIAM Rev 63:435â€“485
Tudisco F, Higham DJ (2021) Node and edge nonlinear eigenvector centrality for hypergraphs. Commun Phys 4:1â€“10
Tudisco F, Higham DJ (2023) Coreperiphery detection in hypergraphs. SIAM J Math Data Sci 5:1â€“21
Vasilyeva E et al (2021) Multilayer representation of collaboration networks with higherorder interactions. Sci Rep 11:5666
Vazquez A (2009) Finding hypergraph communities: a bayesian approach and variational solution. J Stat Mech Theory Exp 2009:P07006
Wolf MM, Klinvex AM, Dunlavy DM (2016) Advantages to modeling relational data using hypergraphs versus graphs. In: 2016 IEEE high performance extreme computing conference (HPEC), pp 1â€“7
Young JG, Petri G, Peixoto TP (2021) Hypergraph reconstruction from network data. Commun Phys 4:1â€“11
Zhang Y, Lucas M, Battiston F (2023) Higherorder interactions shape collective dynamics differently in hypergraphs and simplicial complexes. Nat Commun 14:1605
Acknowledgements
F.B. acknowledges support from the Air Force Office of Scientific Research under award number FA86552217025. A.M. acknowledges support from the European Union through Horizon Europe CLOUDSTARS project (101086248).
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Lotito, Q.F., Montresor, A. & Battiston, F. Multiplex measures for higherorder networks. Appl Netw Sci 9, 55 (2024). https://doi.org/10.1007/s41109024006659
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DOI: https://doi.org/10.1007/s41109024006659