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Schematic illustration of the co-authorship hypergraph (a) and of the dual hypergraph (b). In panel (a) nodes are authors, and hyperlinks are co-authored Manuscript. The hyperlinks are labeled with letters and colours. The legend at the bottom of the Figure reports for each letter the corresponding Manuscript’s identifier in the ArXiv. In the legend, moreover, Manuscripts are grouped in coloured boxes, and different colours stand for a different number of coauthors: yellow papers are authored by a single Scholar, whereas green, red and blue Manuscripts are co-authored by two, three and four Scholars, respectively. Panel (b) contains a sketch of the dual representation, where nodes are now papers [labeled with the same colours and letters than in panel (a)], and links are labeled with the name of the authors who participated in the co-authorship of the Manuscripts.

Schematic illustration of the co-authorship hypergraph (a) and of the dual hypergraph (b). In panel (a) nodes are authors, and hyperlinks are co-authored Manuscript. The hyperlinks are labeled with letters and colours. The legend at the bottom of the Figure reports for each letter the corresponding Manuscript’s identifier in the ArXiv. In the legend, moreover, Manuscripts are grouped in coloured boxes, and different colours stand for a different number of coauthors: yellow papers are authored by a single Scholar, whereas green, red and blue Manuscripts are co-authored by two, three and four Scholars, respectively. Panel (b) contains a sketch of the dual representation, where nodes are now papers [labeled with the same colours and letters than in panel (a)], and links are labeled with the name of the authors who participated in the co-authorship of the Manuscripts.

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Collaboration patterns offer important insights into how scientific breakthroughs and innovations emerge in small and large research groups. However, links in traditional networks account only for pairwise interactions, thus making the framework best suited for the description of two-person collaborations, but not for collaborations in larger group...

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... An example of a network that captures higher-order interactions between different entity types is the bipartite network, where nodes are partitioned by two separate groups, and edges only connect nodes from different groups. Bipartite networks are particularly suitable for modeling systems where two types of entities interact, such as authors and papers in a collaboration network [18], recommendation systems [19] where nodes represent users and the recommending items, and specifically in ...
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Analysis of single-cell RNA sequencing data is often conducted through network projections such as coexpression networks, primarily due to the abundant availability of network analysis tools for downstream tasks. However, this approach has several limitations: loss of higher-order information, inefficient data representation caused by converting a sparse dataset to a fully connected network, and overestimation of coexpression due to zero-inflation. To address these limitations, we propose conceptualizing scRNA-seq expression data as hypergraphs, which are generalized graphs in which the hyperedges can connect more than two vertices. In the context of scRNA-seq data, the hypergraph nodes represent cells and the edges represent genes. Each hyperedge connects all cells where its corresponding gene is actively expressed and records the expression of the gene across different cells. This hypergraph conceptualization enables us to explore multi-way relationships beyond the pairwise interactions in coexpression networks without loss of information. We propose two novel clustering methods: (1) the Dual-Importance Preference Hypergraph Walk (DIPHW) and (2) the Coexpression and Memory-Integrated Dual-Importance Preference Hypergraph Walk (CoMem-DIPHW). They outperform established methods on both simulated and real scRNA-seq datasets. The improvement brought by our proposed methods is especially significant when data modularity is weak. Furthermore, CoMem-DIPHW incorporates the gene coexpression network, cell coexpression network, and the cell-gene expression hypergraph from the single-cell abundance counts data altogether for embedding computation. This approach accounts for both the local level information from single-cell level gene expression and the global level information from the pairwise similarity in the two coexpression networks.
... In systems composed of multiple particles, interactions may go beyond pairwise relations and involve the collective action of groups of agents that cannot be decomposed. A classic example is collaboration networks, where more than two people can participate in a project or coauthor a paper [1]. In physics, the Einstein-Infeld-Hoffmann equations of motion, which incorporate small general-relativistic effects into many-body newtonian mechanics, lead to gravitational forces that are proportional to the product of several different masses [2,3]. ...
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Higher order interactions can lead to new equilibrium states and bifurcations in systems of coupled oscillators described by the Kuramoto model. However, even in the simplest case of 3-body interactions there are more than one possible functional forms, depending on how exactly the bodies are coupled. Which of these forms is better suited to describe the dynamics of the oscillators depends on the specific system under consideration. Here we show that, for a particular class of interactions, reduced equations for the Kuramoto order parameter can be derived for arbitrarily many bodies. Moreover, the contribution of a given term to the reduced equation does not depend on its order, but on a certain effective order, that we define. We give explicit examples where bi and tri-stability is found and discuss a few exotic cases where synchronization happens via a third order phase transition.
... Also in case of disease infection, one healthy unit can be infected in touch with multiple infected units, and takes the form of higher-order interactions [42,43]. One can also take the example of a collaboration network, where the dynamics of multiauthor collaboration can not be described by the combination of pairwise collaborations [44]. In such systems, the multiauthor interactions can be represented as higher-order interactions which cannot always be expressed as a sum of pairwise interactions. ...
... Now it can be easily checked that λ = 0 is the only solution of Eqs. (44) and (45) produces the solution 2λ = K 1 r a 1 a + K 2 r b+2 ...
... There, the contribution of links of different orders can be indicative of how smaller or larger author teams contribute to the connectivity and integration of global science. [52][53][54] First methodological steps towards determining the relevance of the ties of different orders were taken by Vasilyeva et al., 55 who proposed a multi-layer network representation to identify the smallest size of group interactions that contribute significantly to the network structure. Recently, a filtering procedure to remove small, large or specific orders was proposed by Landry et al., 56 to investigate how particular global and local network properties are affected when specific orders are preserved or filtered out. ...
... Moreover, the precise contribution of different orders to the network structure could vary depending on which topological network measure is considered. 55 This underlines the need of investigating the contribution of hyperlinks with different sizes to the network structure at different levels of analysis. One can focus on the contribution of different orders to the structure of the traditional pairwise (or projected) representation of the network, where each pair of nodes is connected by a link if they are connected by a hyperlink of any order. ...
... . Hypergraph H − d is equivalent to the one proposed by Vasilyeva et al. 55 and obtained by the lower or equal (LEQ) filtering of Landry et al. 56 Differently, H + d is obtained by including only hyperlinks with order d ′ > d, i.e., ...
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Higher-order networks effectively represent complex systems with group interactions. Existing methods usually overlook the relative contribution of group interactions (hyperlinks) of different sizes to the overall network structure. Yet, this has many important applications, especially when the network has meaningful node labels. In this work, we propose a comprehensive methodology to precisely measure the contribution of different orders to the overall network structure. First, we propose the order contribution measure, which quantifies the contribution of hyperlinks of different orders to the link weights (local scale), number of triangles (mesoscale) and size of the largest connected component (global scale) of the pairwise weighted network. Second, we propose the measure of order relevance, which gives insights in how hyperlinks of different orders contribute to the considered network property. Most interestingly, it enables an assessment of whether this contribution is synergistic or redundant with respect to that of hyperlinks of other orders. Third, to account for labels, we propose a metric of label group balance to assess how hyperlinks of different orders connect label-induced groups of nodes. We applied these metrics to a large-scale board interlock network and scientific collaboration network, in which node labels correspond to geographical location of the nodes. Experiments including a comparison with randomized null models reveal how from the global level perspective, we observe synergistic contributions of orders in the board interlock network, whereas in the collaboration network there is more redundancy. The findings shed new light on social scientific debates on the role of busy directors in global business networks and the connective effects of large author teams in scientific collaboration networks.
... Similarly, Ref. 37 investigates higher-order interactions in a memristive Rulkov model network, using master stability functions to analyze synchronization patterns, and demonstrates that incorporating higher-order interactions lowers the required coupling parameters for synchronization while also showing that larger network sizes enhance synchronization dynamics and facilitate cluster synchronization under specific coupling conditions. Many other intriguing studies on higher-order interactions [38][39][40][41][42][43][44][45][46][47][48] exist; however, most of them primarily emphasis on long-term behaviors. ...
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Understanding how species interactions shape biodiversity is a core challenge in ecology. While much focus has been on long-term stability, there is rising interest in transient dynamics—the short-lived periods when ecosystems respond to disturbances and adjust toward stability. These transitions are crucial for predicting ecosystem reactions and guiding effective conservation. Our study introduces a model that uses convex combinations to blend pairwise and higher-order interactions (HOIs), offering a more realistic view of natural ecosystems. We find that pairwise interactions slow the journey to stability, while HOIs speed it up. Employing global stability analysis and numerical simulations, we establish that as the proportion of HOIs increases, mean transient times exhibit a significant reduction, thereby underscoring the essential role of HOIs in enhancing biodiversity stabilization. Our results reveal a robust correlation between the most negative real part of the eigenvalues of the Jacobian matrix associated with the linearized system at the coexistence equilibrium and the mean transient times. This indicates that a more negative leading eigenvalue correlates with accelerated convergence to stable coexistence abundances. This insight is vital for comprehending ecosystem resilience and recovery, emphasizing the key role of HOIs in promoting stabilization. Amid growing interest in transient dynamics and its implications for biodiversity and ecological stability, our study enhances the understanding of how species interactions affect both transient and long-term ecosystem behavior. By addressing a critical gap in ecological theory and offering a practical framework for ecosystem management, our work advances knowledge of transient dynamics, ultimately informing effective conservation strategies.
... Similarly, Ref. [37] investigates higher-order interactions in a memristive Rulkov model network, using master stability functions to analyze synchronization patterns, and demonstrates that incorporating higher-order interactions lowers the required coupling parameters for synchronization while also showing that larger network sizes enhance synchronization dynamics and facilitate cluster synchronization under specific coupling conditions. Many other intriguing studies on higher-order interactions [38][39][40][41][42][43][44][45][46][47][48] exist; however, most of them primarily emphasis on long-term behaviors. ...
Preprint
Full-text available
Understanding how species interactions shape biodiversity is a core challenge in ecology. While much focus has been on long-term stability, there is rising interest in transient dynamics-the short-lived periods when ecosystems respond to disturbances and adjust toward stability. These transitions are crucial for predicting ecosystem reactions and guiding effective conservation. Our study introduces a model that blends pairwise and higher-order interactions, offering a more realistic view of natural ecosystems. We find pairwise interactions slow the journey to stability, while higher-order interactions speed it up. This model provides fresh insights into ecosystem resilience and recovery, helping improve strategies for managing species and ecological disruptions.
... 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;Kivelä 2014) (Vasilyeva 2021) and a significant potential, however, multiplex hypergraphs remain relatively unexplored, and a general set of tools for their analysis is still missing. ...
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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 higher-order 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 system’s mesoscale. We validate our measures with three real-world datasets: scientific co-authorship 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 real-world systems marked by both multiplex and higher-order interactions.
... While pairwise interactions, such as interlayer and intralayer links, are foundational and have yielded valuable insights, real-world systems often involve more intricate relationships. Indeed, systems in the real world, spanning human communications in social networks to neuronal interactions in the brain, can be accurately depicted through multilayer networks where interactions frequently occur among groups of three or more individuals simultaneously [10][11][12][13]. For example, in neuronal networks, neurons are interconnected through electrical and chemical synapses, giving rise to a multilayer structure [8,14]. ...
Article
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Networks incorporating higher-order interactions are increasingly recognized for their ability to introduce novel dynamics into various processes, including synchronization. Previous studies on synchronization within multilayer networks have often been limited to specific models, such as the Kuramoto model, or have focused solely on higher-order interactions within individual layers. Here, we present a comprehensive framework for investigating synchronization, particularly global synchronization, in multilayer networks with higher-order interactions. Our framework considers interactions beyond pairwise connections, both within and across layers. We demonstrate the existence of a stable global synchronous state, with a condition resembling the master stability function, contingent on the choice of coupling functions. Our theoretical findings are supported by simulations using Hindmarsh-Rose neuronal and Rössler oscillators. These simulations illustrate how synchronization is facilitated by higher-order interactions, both within and across layers, highlighting the advantages over scenarios involving interactions within single layers. Published by the American Physical Society 2024
... Also in case of disease infection, one healthy unit can be infected in touch of multiple infected units, takes the form of higher-order interactions [41,42]. One can also take the example of a collaboration network, where the dynamics of multi-author collaboration can not be suffices to describe with the combination of pairwise collaborations [43]. In such systems, the multiauthor interactions can be represented as higher-order interactions which can not always be expressed as a sum of pairwise interactions. ...
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Full-text available
We investigate the phenomenon of transition to synchronization in Sakaguchi-Kuramoto model in the presence of higher-order interactions and global order parameter adaptation. The investigation is done by performing extensive numerical simulations and low dimensional modeling of the system. Numerical simulations of the full system show both continuous (second order) as well as discontinuous transitions. The discontinuous transitions can either be associated with explosive (first order) or with tiered synchronization states depending on the choice of parameters. To develop an in depth understanding of the transition scenario in the parameter space we derive a reduced order model (ROM) using the Ott-Antonsen ansatz, the results of which closely matches with that of the numerical simulations of the full system. The simplicity and analytical accessibility of the ROM helps to conveniently unfold the transition scenario in the system having complex dependence on the parameters. Simultaneous analysis of the full system and the ROM clearly identifies the regions of the parameter space exhibiting different types of transitions. It is observed that the second order continuous transition is connected with a supercritical pitchfork bifurcation (PB) of the ROM. On the other hand, the discontinuous teired transition is associated with multiple saddle-node (SN) bifurcations along with a supercritical PB and the first order explosive transition involves a subcritical PB alongside a SN bifurcation.
... While pairwise interactions, such as interlayer and intralayer links, are foundational and have yielded valuable insights, real-world systems often involve more intricate relationships. Indeed, systems in the real world, spanning human communications in social networks to neuronal interactions in the brain, can be accurately depicted through multilayer networks where interactions frequently occur among groups of three or more individuals simultaneously [10][11][12][13]. For example, in neuronal networks, neurons are interconnected through electrical and * dibakar@isical.ac.in chemical synapses, giving rise to a multilayer structure [8,14]. ...
Preprint
Full-text available
Networks incorporating higher-order interactions are increasingly recognized for their ability to introduce novel dynamics into various processes, including synchronization. Previous studies on synchronization within multilayer networks have often been limited to specific models, such as the Kuramoto model, or have focused solely on higher-order interactions within individual layers. Here, we present a comprehensive framework for investigating synchronization, particularly global synchronization, in multilayer networks with higher-order interactions. Our framework considers interactions beyond pairwise connections, both within and across layers. We demonstrate the existence of a stable global synchronous state, with a condition resembling the master stability function, contingent on the choice of coupling functions. Our theoretical findings are supported by simulations using Hindmarsh-Rose neuronal and R\"{o}ssler oscillators. These simulations illustrate how synchronization is facilitated by higher-order interactions, both within and across layers, highlighting the advantages over scenarios involving interactions within single layers.