High-order Networks Research Articles

Hyper-Null Models and Their Applications.

Null models are crucial tools for investigating network topological structures. However, research on null models for higher-order networks is still relatively scarce. In this study, we introduce an innovative method to construct null models for hypergraphs, namely the hyperedge swapping-based method. By preserving certain network properties while altering others, we generate six hyper-null models with various orders and analyze their interrelationships. To validate our approach, we first employ hypergraph entropy to assess the randomness of these null models across four datasets. Furthermore, we examine the differences in important statistical properties between the various null models and the original networks. Lastly, we investigate the impact of hypergraph randomness on network dynamics using the proposed hyper-null models, focusing on dismantling and epidemic contagion. The findings show that our proposed hyper-null models are applicable to various scenarios. By introducing a comprehensive framework for generating and analyzing hyper-null models, this research opens up avenues for further exploration of the intricacies of network structures and their real-world implications.

Blind super-resolution model based on unsupervised degenerate indication learning

Blind super-resolution of images based on contrast learning has achieved better performance, which can distinguish between different degraded information. However, there is no significant improvement in the processing of image super-resolution. To further improve the performance of image blind super-resolution, a new degenerate higher-order attention network (DHAN) is designed for image super-resolution, in which a higher-order channel attention mechanism is introduced to the framework. Meanwhile, it also introduces normalized variance and global covariance pooling, which can provide higher-order feature information and more discriminative feature information. Comparing the experimental results, it can be found that the proposed higher-order attention network (DHAN) is able to distinguish various complex degradation patterns and thus obtain accurate degradation information. Numerous experiments have shown that higher-order attention network (DHAN) achieves good visual results in the image blind super-resolution task.

An Introduction to Complex Networks in Climate Finance.

In this perspective, we introduce recent research into the structure and function of complex investor networks supporting sustainability efforts. Using the case of solar, wind and hydro energy technologies, this perspective explores the complexity in low-carbon finance markets, defined as markets that direct capital flows towards low-carbon technologies, using network approaches to study their structure and dynamics. Investors are modeled as nodes which form a network or higher-order network connected by edges representing projects in which joint funding or security-related insurance was provided or other investment-related interaction occurred. We review the literature on investor networks generally, particularly in the case of complex networks, and address areas where these ideas were applied in this emerging field. The complex investor dynamics which emerge from the extant funding scenarios are not well understood. These dynamics have the potential to result in interesting non-linear behaviour, growth, and decline, which can be studied, explained and controlled using the tools of network science.

Functional Connectome Hierarchy Distortions in Female Nurses With Occupational Burnout and Its Gene Expression Signatures.

Burnout has become a serious public health issue worldwide, particularly during the COVID-19 pandemic. Functional connectome impairments associated with occupational burnout were widely distributed, involving both low-level sensorimotor cortices and high-level association cortices. To investigate whether there are hierarchical perturbations in the functional connectomes and if these perturbations are potentially influenced by genetic factors in nurses who feel "burned out." Prospective, case control. Thirty-three female nurses with occupational burnout (aged 27-40, 32.42 ± 3.37) and 32 matched nurses who were not feeling burned out (aged 27-42, 32.50 ± 4.21). 3.0 T, gradient-echo echo-planar imaging sequence (GE-EPI). Gradient-based techniques were used to depict the perturbations in the multi-dimensional hierarchical structure of the macroscale connectome. Gene expression data were acquired from the Allen Human Brain Atlas. Cortex-wide multivariate analyses were used for between-group differences in gradients as well as association analyses between the hierarchy distortions and the MBI score (FDR corrected). Partial least squares, spin test and bootstrapping were utilized together to select the gene sets (FDR corrected). Gene enrichment analyses (GO, KEGG and cell-type) were further performed. Significance level: P < 0.05. There were significant gradient distortions, with strong between-group effects in the somatosensory network and moderate effects in the higher-order default-mode network, which were significantly correlated with the gene expression profiles (r = 0.3171). The most related genes were broadly involved in the cellular response to minerals, neuronal plasticity, and the circadian rhythm pathway (q value < 0.01). Significant enrichments were found in excitatory (r = 0.2588), inhibitory neurons (r = 0.2610), and astrocytes cells (r = 0.2633). Regions affected by burnout severity were mainly distributed in the association and visual cortices. By connecting in vivo imaging to genes, cell classes, and clinical data, this study provides a framework to understand functional impairments in occupational burnout and how the microscopic genetic architecture drive macroscopic distortions. 1 TECHNICAL EFFICACY: Stage 2.

Abnormal higher-order network interactions in Parkinson's disease visual hallucinations.

Visual hallucinations in Parkinson's disease can be viewed from a systems-level perspective, whereby dysfunctional communication between brain networks responsible for perception predisposes a person to hallucinate. To this end, abnormal functional interactions between higher-order and primary sensory networks have been implicated in the pathophysiology of visual hallucinations in Parkinson's disease, however the precise signatures remain to be determined. Dimensionality reduction techniques offer a novel means for simplifying the interpretation of multidimensional brain imaging data, identifying hierarchical patterns in the data that are driven by both within- and between-functional network changes. Here, we applied two complementary non-linear dimensionality reduction techniques-diffusion-map embedding and t-distributed stochastic neighbour embedding (t-SNE)-to resting state functional MRI data, in order to characterize the altered functional hierarchy associated with susceptibility to visual hallucinations. Our study involved 77 people with Parkinson's disease (31 with hallucinations; 46 without hallucinations) and 19 age-matched healthy control subjects. In patients with visual hallucinations, we found compression of the unimodal-heteromodal gradient consistent with increased functional integration between sensory and higher order networks. This was mirrored in a traditional functional connectivity analysis, which showed increased connectivity between the visual and default mode networks in the hallucinating group. Together, these results suggest a route by which higher-order regions may have excessive influence over earlier sensory processes, as proposed by theoretical models of hallucinations across disorders. By contrast, the t-SNE analysis identified distinct alterations in prefrontal regions, suggesting an additional layer of complexity in the functional brain network abnormalities implicated in hallucinations, which was not apparent in traditional functional connectivity analyses. Together, the results confirm abnormal brain organization associated with the hallucinating phenotype in Parkinson's disease and highlight the utility of applying convergent dimensionality reduction techniques to investigate complex clinical symptoms. In addition, the patterns we describe in Parkinson's disease converge with those seen in other conditions, suggesting that reduced hierarchical differentiation across sensory-perceptual systems may be a common transdiagnostic vulnerability in neuropsychiatric disorders with perceptual disturbances.

Some generalized centralities in higher-order networks represented by simplicial complexes

Abstract Higher-order interactions, that is, interactions among the units of group size greater than two, are a fundamental structural feature of a variety of complex systems across the scale. Simplicial complexes are combinatorial objects that can capture and model the higher-order interactions present in a given complex system and thus represent the complex system as a higher-order network comprising simplices. In this work, a given simplicial complex is viewed as a finite union of d-exclusive simplicial complexes. Thus, to represent a complex system as a higher-order network given by a simplicial complex that captures all orders of interactions present in the system, a family of symmetric adjacency tensors A(d) of dimension d + 1 and appropriate order has been used. Each adjacency tensor A(d) represents a d-exclusive simplicial complex and for d≥2 it represents exclusively higher-order interactions of the system. For characterizing the structure of d-exclusive simplicial complexes, the notion of generalized structural centrality indices namely, generalized betweenness centrality and generalized closeness centrality has been established by developing the concepts of generalized walk and generalized distance in the simplicial complex. Generalized centrality indices quantify the contribution of δ-simplices in any d-exclusive simplicial complex Δ, where δ&amp;lt;d and if d≥2, it describes the contribution of δ-faces to the higher-order interactions of Δ. These generalized centrality indices provide local structural descriptions, which lead to mesoscale insights into the simplicial complex that comprises the higher-order network. An important theorem providing a general technique for the characterization of connectedness in d-exclusive simplicial complexes in terms of irreducibility of its adjacency tensor has been established. The concepts developed in this work together with concepts of generalized simplex deletion in d-exclusive simplicial complexes have been illustrated using examples. The effect of deletions on the generalized centralities of the complexes in the examples has been discussed.

A Graph Representation Learning Framework Predicting Potential Multivariate Interactions

Link prediction is a widely adopted method for extracting valuable data insights from graphs, primarily aimed at predicting interactions between two nodes. However, there are not only pairwise interactions but also multivariate interactions in real life. For example, reactions between multiple proteins, multiple compounds, and multiple metabolites cannot be mined effectively using link prediction. A hypergraph is a higher-order network composed of nodes and hyperedges, where hyperedges can be composed of multiple nodes, and can be used to depict multivariate interactions. The interactions between multiple nodes can be predicted by hyperlink prediction methods. Since hyperlink prediction requires predicting the interactions between multiple nodes, it makes the study of hyperlink prediction much more complicated than that of other complex networks, thus resulting in relatively limited attention being devoted to this field. The existing hyperlink prediction can only predict potential hyperlinks in uniform hypergraphs, or need to predict hyperlinks based on the candidate hyperlink sets, or only study hyperlink prediction for undirected hypergraphs. Therefore, a hyperlink prediction framework for predicting multivariate interactions based on graph representation learning is proposed to solve the above problems, and then the framework is extended to directed hyperlink prediction (e.g., directed metabolic reaction networks). Furthermore, any size of hyperedges can be predicted by the proposed hyperlink prediction algorithm framework, whose performance is not affected by the number of nodes or the number of hyperedges. Finally, the proposed framework is applied to both the biological metabolic reaction network and the organic chemical reaction network, and experimental analysis has demonstrated that the hyperlinks can be predicted efficiently by the proposed hyperlink prediction framework with relatively low time complexity, and the prediction performance has been improved by up to 40% compared with the baselines.

HYPERMATRIX ALGEBRA AND IRREDUCIBLE ARITY IN HIGHER-ORDER SYSTEMS: CONCEPTS AND PERSPECTIVES

Theoretical and computational frameworks of complexity science are dominated by binary structures. This binary bias, seen in the ubiquity of pair-wise networks and formal binary operations in mathematical models, limits our capacity to faithfully capture irreducible polyadic interactions in higher-order systems. A paradigmatic example of a higher-order interaction is the Borromean link of three interlocking rings. In this paper, we propose a mathematical framework via hypergraphs and hypermatrix algebras that allows to formalize such forms of higher-order bonding and connectivity in a parsimonious way. Our framework builds on and extends current techniques in higher-order networks — still mostly rooted in binary structures such as adjacency matrices — and incorporates recent developments in higher-arity structures to articulate the compositional behavior of adjacency hypermatrices. Irreducible higher-order interactions turn out to be a widespread occurrence across natural sciences and socio-cultural knowledge representation. We demonstrate this by reviewing recent results in computer science, physics, chemistry, biology, ecology, social science, and cultural analysis through the conceptual lens of irreducible higher-order interactions. We further speculate that the general phenomenon of emergence in complex systems may be characterized by spatio-temporal discrepancies of interaction arity.

Dirac signal processing of higher-order topological signals

Higher-order networks can sustain topological signals which are variables associated not only to the nodes, but also to the links, to the triangles and in general to the higher dimensional simplices of simplicial complexes. These topological signals can describe a large variety of real systems including currents in the ocean, synaptic currents between neurons and biological transportation networks. In real scenarios topological signal data might be noisy and an important task is to process these signals by improving their signal to noise ratio. So far topological signals are typically processed independently of each other. For instance, node signals are processed independently of link signals, and algorithms that can enforce a consistent processing of topological signals across different dimensions are largely lacking. Here we propose Dirac signal processing, an adaptive, unsupervised signal processing algorithm that learns to jointly filter topological signals supported on nodes, links and triangles of simplicial complexes in a consistent way. The proposed Dirac signal processing algorithm is formulated in terms of the discrete Dirac operator which can be interpreted as ‘square root’ of a higher-order Hodge Laplacian. We discuss in detail the properties of the Dirac operator including its spectrum and the chirality of its eigenvectors and we adopt this operator to formulate Dirac signal processing that can filter noisy signals defined on nodes, links and triangles of simplicial complexes. We test our algorithms on noisy synthetic data and noisy data of drifters in the ocean and find that the algorithm can learn to efficiently reconstruct the true signals outperforming algorithms based exclusively on the Hodge Laplacian.

Synchronization in simplicial complexes of memristive Rulkov neurons.

Simplicial complexes are mathematical constructions that describe higher-order interactions within the interconnecting elements of a network. Such higher-order interactions become increasingly significant in neuronal networks since biological backgrounds and previous outcomes back them. In light of this, the current research explores a higher-order network of the memristive Rulkov model. To that end, the master stability functions are used to evaluate the synchronization of a network with pure pairwise hybrid (electrical and chemical) synapses alongside a network with two-node electrical and multi-node chemical connections. The findings provide good insight into the impact of incorporating higher-order interaction in a network. Compared to two-node chemical synapses, higher-order interactions adjust the synchronization patterns to lower multi-node chemical coupling parameter values. Furthermore, the effect of altering higher-order coupling parameter value on the dynamics of neurons in the synchronization state is researched. It is also shown how increasing network size can enhance synchronization by lowering the value of coupling parameters whereby synchronization occurs. Except for complete synchronization, cluster synchronization is detected for higher electrical coupling strength values wherein the neurons are out of the completed synchronization state.

Diffusibility of a kind of quasi-star higher-order network

Real life networks heavily rely on higher-order interactions. This paper studies the diffusibility of novel quasi-star higher-order networks from two aspects. When the coupling strengths of the lower-order and higher-order coupling parts are less than 1, the diffusibility can be maximized by choosing the intermediate values of these coupling strengths. When they are far greater than 1, there exists an inferior limit of the diffusibility, and the lower-order coupling part has more impact on quasi-star network’s diffusibility. Additionally, the face with the middle order in the higher-order coupling part plays a more important role in enhancing the network’s diffusibility compared with other faces, and the diffusibility does not naturally increase with the order of face. These results give us some insights into the influence of higher-order structures and their faces.

Influence maximization in hypergraphs: A self-optimizing algorithm based on electrostatic field

Multi-individual interactions are ubiquitous in the real world, which are usually modeled by hypergraphs. Similar to low-order networks, there are special key nodes in high-order networks that are highly influential and play a crucial role in information dissemination. The objective of the influence maximization problem is to find an optimal set of nodes that maximizes the influence of the network, while maximizing influence in hypergraphs has been neglected so far, and how to explore higher-order interactions from multiple perspectives and exploit the features of higher-order structures are still open questions. To solve the above problems, a self-optimization algorithm based on electrostatic field is proposed. The network is innovatively viewed as an electrostatic field to uncover the interaction forces of the nodes in it. The improved localized Physarum polycephalum algorithm is used to assign values to the initial charges of the nodes. The effective distance among hyperedges and nodes is redefined. Furthermore, the self-optimizing update algorithm is designed to reduce the overlap of nodes’ influence ranges in the optimal set. Percolation, SIR, Top-k, and correlation experiments are conducted on eight real-world networks, and the superiority of our proposed algorithm in this paper is verified by comparing with six algorithms.

Robustness of directed higher-order networks.

In complex systems, from human social networks to biological networks, pairwise interactions are insufficient to express the directed interactions in higher-order networks since the internal function is not only contained in directed pairwise interactions but rather in directed higher-order interactions. Therefore, researchers adopted directed higher-order networks to encode multinode interactions explicitly and revealed that higher-order interactions induced rich critical phenomena. However, the robustness of the directed higher-order networks has yet to receive much attention. Here, we propose a theoretical percolation model to analyze the robustness of directed higher-order networks. We study the size of the giant connected components and the percolation threshold of our proposed model by the theory and Monte-Carlo simulations on artificial networks and real-world networks. We find that the percolation threshold is affected by the inherent properties of higher-order networks, including the heterogeneity of the hyperdegree distribution and the hyperedge cardinality, which represents the number of nodes in the hyperedge. Increasing the hyperdegree distribution of heterogeneity or the hyperedge cardinality distribution of heterogeneity in higher-order networks will make the network more vulnerable, weakening the higher-order network's robustness. In other words, adding higher-order directed edges enhances the robustness of the systems. Our proposed theory can reasonably predict the simulations for percolation on artificial and real-world directed higher-order networks.

What Are Higher-Order Networks?

What Are Higher-Order Networks?

Higher-order link prediction via local information.

Link prediction has been widely studied as an important research direction. Higher-order link prediction has gained, in particular, significant attention since higher-order networks provide a more accurate description of real-world complex systems. However, higher-order networks contain more complex information than traditional pairwise networks, making the prediction of higher-order links a formidable challenging task. Recently, researchers have discovered that local features have advantages over long-range features in higher-order link prediction. Therefore, it is necessary to develop more efficient and concise higher-order link prediction algorithms based on local features. In this paper, we proposed two similarity metrics via local information, simplicial decomposition weight and closed ratio weight, to predict possible future higher-order interactions (simplices) in simplicial networks. These two algorithms capture local higher-order information at two aspects: simplex decomposition and cliques' state (closed or open). We tested their performance in eight empirical simplicial networks, and the results show that our proposed metrics outperform other benchmarks in predicting third-order and fourth-order interactions (simplices) in most cases. In addition, we explore the robustness of the proposed algorithms, and the results suggest that the performance of these novel algorithms is advanced under different sizes of training sets.

A systematic scoping review of rodent models of catatonia: Clinical correlations, translation and future approaches

Catatonia is a psychiatric disorder, which subsumes a plethora of affective, motor and behavioral symptoms. In the last two decades, the number of behavioral and neuroimaging studies on catatonia has steadily increased. The majority of behavioral and neuroimaging studies in psychiatric patients suggested aberrant higher-order frontoparietal networks which, on the biochemical level, are insufficiently modulated by gamma-aminobutyric acid (GABA)-ergic and glutamatergic transmission. However, the pathomechanisms of catatonic symptoms have rarely been studied using rodent models. Here, we performed a scoping review of literature available on PubMed for studies on rodent models of catatonia. We sought to identify what we could learn from pre-clinical animal models of catatonia-like symptoms, their underlying neuronal correlates, and the complex molecular (i.e. genes and neurotransmitter) mechanisms by which its modulation exerts its effects. What becomes evident is that although many transgenic models present catatonia-like symptoms, they have not been used to better understand the pathophysiological mechanisms underlying catatonia so far. However, the identified neuronal correlates of catatonia-like symptoms correlate to a great extent with findings from neuroscience research in psychiatric patients. This points us towards fundamental cortical-striatal-thalamocortical and associated networks modulated by white matter inflammation as well as aberrant dopaminergic, GABAergic, and glutamatergic neurotransmission that is involved in catatonia. Therefore, this scoping review opens up the possibility of finally using transgenic models to help with identifying novel target mechanisms for the development of new drugs for the treatment of catatonia.

Heterogeneous graph attention network with motif clique

Heterogeneous information networks (HINs) have been proven to be powerful in modeling various real-world networks, such as academic networks and social media networks. By distinguishing the types of relationships and nodes, HINs obtain the capability to model multifarious data, while bringing challenges to data mining and analysis. Specifically, how to comprehensively mine the semantic information in HINs remains an open question. Although recently studied network schemas such as meta-paths or meta-graphs have achieved empirical success, they still have limitations in modeling the complex underlying relations to facilitate the graph neural networks, e.g., different schemas may overlap in semantics and lead to redundant computations. To address this issue, we leverage a recently introduced higher-order network schema known as the motif clique (abbreviated as m-clique). This schema offers greater expressiveness, enabling us to effectively construct the semantic neighborhood of nodes. We further propose a novel heterogeneous graph attention network with m-cliques, named HAMC, which employs a two-level attention mechanism (node-level and semantic-level) to learn node representations. The two-level attention measures the importance of neighbors and m-cliques for each node, respectively. Extensive experiments demonstrate that the proposed HAMC outperforms the state-of-the-art methods in many heterogeneous network analytic tasks such as node classification and clustering. Our code is available at https://github.com/wcx21/HAMC-Heterogeneous-Graph-Attention-Network-with-Motif-Clique.

Research on User Behavior Based on Higher-Order Dependency Network.

In the era of the popularization of the Internet of Things (IOT), analyzing people's daily life behavior through the data collected by devices is an important method to mine potential daily requirements. The network method is an important means to analyze the relationship between people's daily behaviors, while the mainstream first-order network (FON) method ignores the high-order dependencies between daily behaviors. A higher-order dependency network (HON) can more accurately mine the requirements by considering higher-order dependencies. Firstly, our work adopts indoor daily behavior sequences obtained by video behavior detection, extracts higher-order dependency rules from behavior sequences, and rewires an HON. Secondly, an HON is used for the RandomWalk algorithm. On this basis, research on vital node identification and community detection is carried out. Finally, results on behavioral datasets show that, compared with FONs, HONs can significantly improve the accuracy of random walk, improve the identification of vital nodes, and we find that a node can belong to multiple communities. Our work improves the performance of user behavior analysis and thus benefits the mining of user requirements, which can be used to personalized recommendations and product improvements, and eventually achieve higher commercial profits.

An Evolutionary Analysis of Higher-Order Interaction Collaborative Innovation Networks in China’s New Energy Vehicle Industry

As a manifestation of technological innovation achievements, patents reflect the frontier of technological development in the field. The aim of this research is to investigate the spatial evolution of patent collaboration networks and cooperation activities in the Chinese new energy vehicle (NEV) industry. We hypothesize that the higher-order collaboration networks will exhibit the formation of triangle groups centered around core nodes and the emergence of key edges indicating their importance. Key organizations and partners will play a significant role in shaping the future direction of cooperative innovation. The research utilizes data on collaboration patents in the NEV industry in China and constructs higher-order interaction collaboration innovation networks. The spatial evolution of networks and patent cooperation activities are examined using simplex methods. The results indicate that the generalized degree distribution of nodes and edges follows a power-law distribution. Two-dimensional simplex networks gradually form triangle clusters centered on core nodes and key edges. Large companies and research institutes prefer high-depth collaboration, while universities prefer high-breadth collaboration. Furthermore, the development of the new energy vehicle industry has gradually shifted from the eastern region to the central region. In the two-dimensional simplex network, Beijing and Jiangsu play a crucial role as key bridges in fostering stable and deep collaborations. The findings of this study shed light on the spatial evolution of patent collaboration networks and cooperation activities in the Chinese NEV industry. The identification of key organizations and partners, as well as the central positions of certain regions, provides valuable insights for innovation organizations in navigating innovation development and selecting collaboration partners.

Mapping change in higher-order networks with multilevel and overlapping communities

New network models of complex systems use layers, state nodes, or hyperedges to capture higher-order interactions and dynamics. Simplifying how the higher-order networks change over time or depending on the network model would be easy with alluvial diagrams, which visualize community splits and merges between networks. However, alluvial diagrams were developed for networks with regular nodes assigned to non-overlapping flat communities. How should they be defined for nodes in layers, state nodes, or hyperedges? How can they depict multilevel, overlapping communities? Here we generalize alluvial diagrams to map change in higher-order networks and provide an interactive tool for anyone to generate alluvial diagrams. We use the alluvial diagram generator in three case studies to illustrate significant changes in the organization of science, the effect of modeling network flows with memory in a citation network and distinguishing multidisciplinary from field-specific journals, and the effects of multilayer representation of a collaboration hypergraph.