Higher-order Interactions Research Articles

Identification of important nodes in multi-layer hypergraphs based on fuzzy gravity model and node centrality distribution characteristics

Hyperedge is a common structure that represents high-order interactions between nodes in complex networks, and multi-layer networks provide more diverse node interactions than single-layer networks. Therefore, multi-layer hypergraphs can more clearly represent the relationships between nodes. However, there are few studies on identifying important nodes in this framework. This paper proposes a method called HCT to fill this gap. It consists of three parts, namely: Hypergraph Fuzzy Gravity Model (HFGM), Layer Weight Calculation Method based on Node Centrality Distribution Characteristics (CDLW) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). HCT progressively analyzes the importance of a node from three levels: local (the node itself), semi-local (the hyperedges and community to which the node belongs), and global (the layer to which the node belongs). By combining these three results, the global centrality of a node in the entire network can be calculated. Simulation experiments demonstrate that the important nodes identified by HCT exhibit stronger contagion capabilities in nine networks compared to nine combinatorial methods and removing these nodes will seriously damage the connectivity and robustness of the network. The centrality of each node calculated by HCT is also consistent with its actual importance.

The simplicity of protein sequence-function relationships

How complex are the rules by which a protein’s sequence determines its function? High-order epistatic interactions among residues are thought to be pervasive, suggesting an idiosyncratic and unpredictable sequence-function relationship. But many prior studies may have overestimated epistasis, because they analyzed sequence-function relationships relative to a single reference sequence—which causes measurement noise and local idiosyncrasies to snowball into high-order epistasis—or they did not fully account for global nonlinearities. Here we present a reference-free method that jointly infers specific epistatic interactions and global nonlinearity using a bird’s-eye view of sequence space. This technique yields the simplest explanation of sequence-function relationships and is more robust than existing methods to measurement noise, missing data, and model misspecification. We reanalyze 20 experimental datasets and find that context-independent amino acid effects and pairwise interactions, along with a simple nonlinearity to account for limited dynamic range, explain a median of 96% of phenotypic variance and over 92% in every case. Only a tiny fraction of genotypes are strongly affected by higher-order epistasis. Sequence-function relationships are also sparse: a miniscule fraction of amino acids and interactions account for 90% of phenotypic variance. Sequence-function causality across these datasets is therefore simple, opening the way for tractable approaches to characterize proteins’ genetic architecture.

The Impact of Cardiovascular Disease Gene Polymorphism and Interaction with Homocysteine on Deep Vein Thrombosis.

Deep vein thrombosis (DVT) affects vascular health and can even threaten life; however, its pathogenesis remains unclear. Cardiovascular disease (CVD) and DVT share common risk factors, such as dyslipidemia, aging, etc. We aimed to investigate the loci of published CVD susceptibility genes and their association with environmental factors that might be related to DVT. Genotyping by Kompetitive Allele Specific PCR (KASP), collection of lifestyle information, and determination of blood biochemical markers were performed in 165 DVT cases and 164 controls. The impact of six single nucleotide polymorphisms (SNPs) and additional potential variables on DVT morbidity was evaluated using unconditional logistic regression (ULR). To explore the high-order interactions related to genetics and the body's internal environment exposure that affect DVT, ULR, crossover analysis, and multifactor dimensionality reduction/generalized multifactor dimensionality reduction (MDR/GMDR) were employed. Sensitivity analyses were performed using the EpiR package. The polymorphisms of FGB rs1800790 and PLAT rs2020918 were significantly associated with DVT. The optimum GMDR interaction model for gene-gene (G × G) consisted of THBD rs1042579, PLAT rs2020918, and PON1 rs662. The PLAT rs2020918 and MTHFR rs1801133 polymorphisms together eliminated the maximum entropy by the MDR method. The optimum GMDR interaction model for gene-environment (G × E) consisted of MTHFR rs1801133, FGB rs1800790, PLAT rs2020918, PON1 rs662, and total homocysteine (tHcy). Those with high tHcy levels and three risk genotypes significantly increased the DVT risk. In conclusion, certain CVD-related SNPs and their interactions with tHcy may contribute to DVT. These have implications for investigating DVT etiology and developing preventive treatment plans.

Pattern formation in reaction-diffusion information propagation model on multiplex simplicial complexes

Turing patterns for explaining spatial distribution in nature mostly focus on continuous media and existing networks, but only few attempts exist to study them on systems with high-order interactions. Considering that high-order interactions have a particularly significant impact on rumor propagation, this study establishes a generalized reaction-diffusion rumor propagation model based on a multiplex network, where simplicial complexes are employed to describe high-order structures. It aims to provide the spatial distribution patterns of the population participating in rumor propagation and identify the structural factors that affect such patterns. We theoretically provide the necessary conditions for Turing instability in single-layer and multiplex networks by considering high-order interactions. In the numerical simulation, we demonstrate that the Turing pattern could be controlled by adjusting the diffusion coefficient, high-order structure intensity, and average degree of the network. The results indicate that: (i) in a single-layer network, the Turing pattern only exists when high-order interactions appear, and the difference in diffusion rate plays a decisive role, (ii) in a multiplex network, the Turing pattern can still be observed under the same diffusion rates, which are affected by the difference in higher-order intensity between the two layers, and (iii) in existing networks, the average degree of the network has an important impact on Turing pattern. These findings contribute toward comprehending the impact of network structure on pattern formation, particularly the high-order interactions on the Turing pattern.

Higher-order interaction induced chimeralike state in a bipartite network.

We report higher-order coupling induced stable chimeralike state in a bipartite network of coupled phase oscillators without any time-delay in the coupling. We show that the higher-order interaction breaks the symmetry of the homogeneous synchronized state to facilitate the manifestation of symmetry breaking chimeralike state. In particular, such symmetry breaking manifests only when the pairwise interaction is attractive and higher-order interaction is repulsive, and vice versa. Further, we also demonstrate the increased degree of heterogeneity promotes homogeneous symmetric states in the phase diagram by suppressing the asymmetric chimeralike state. We deduce the low-dimensional evolution equationsfor the macroscopic order parameters using Ott-Antonsen ansatz and obtain the bifurcation curves from them using the software xppaut, which agrees very well with the simulation results. We also deduce the analytical stability conditions for the incoherent state, in-phase and out-of-phase synchronized states, which match with the bifurcation curves.

Modeling and analysis of cascading failures in multilayer higher-order networks

With the increasing application of network science, understanding the behavior of higher-order networks has become particularly important. Especially, the study of multilayer higher-order networks is significant for revealing the interdependence and higher-order interactions in complex systems. This paper proposes a mathematical framework to explore multilayer higher-order networks composed of multiple fully interdependent hypergraphs. Each hypergraph consists of the same number of nodes, and nodes between layers are connected one-to-one. By analyzing the cascading failures of multilayer hypergraphs under different topologies, we found that although the network sizes are the same in the steady state, different topological structures (such as star-like, tree-like, and chain-like) significantly affect the time required to reach a stable state, with the star-like topology reaching stability the fastest. Furthermore, we explored the impact of network parameters on the robustness of networks. We found that in multilayer homogeneous hypergraphs, the robustness of the network becomes stronger with an increase in the average hyperdegree or average hyperedge cardinality; in multilayer heterogeneous hypergraphs, the robustness of the network becomes more fragile as the power exponent increases. Finally, experimental results indicate that with the rise in the number of network layers, the network becomes more fragile.

Dynamical Fluctuations of Random Walks in Higher-Order Networks

Although higher-order interactions are known to affect the typical state of dynamical processes giving rise to new collective behavior, how they drive the emergence of rare events and fluctuations is still an open problem. We investigate how fluctuations of a dynamical quantity of a random walk exploring a higher-order network arise over time. In the quenched case, where the hypergraph structure is fixed, through large deviation theory we show that the appearance of rare events is hampered in nodes with many higher-order interactions, and promoted elsewhere. Dynamical fluctuations are further boosted in an annealed scenario, where both the diffusion process and higher-order interactions evolve in time. Here, extreme fluctuations generated by optimal higher-order configurations can be predicted in the limit of a saddle-point approximation. Our study lays the groundwork for a wide and general theory of fluctuations and rare events in higher-order networks. Published by the American Physical Society 2024

Beyond directed hypergraphs: heterogeneous hypergraphs and spectral centralities

Abstract The study of hypergraphs has received a lot of attention over the past few years, however up until recently there has been no interest in systems where higher order interactions are not undirected. In this article, we introduce the notion of heterogeneous hypergraphs from an algebraic point of view, which have traditional directed hypergraphs as a particular case. We furthermore analytically study the spectral centralities associated to some types of heterogeneous hypergraphs, extending previously defined eigenvector-like centrality measures to this new realm. We supplement the analytical arguments with some numerical comparisons of pairwise and higher order rankings, and we construct directed higher order networks from real data, which we then use for discussion and analysis.

Multiplex measures for higher-order networks

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.

Assessing the Limitations of Relief-Based Algorithms in Detecting Higher-Order Interactions.

The investigation of epistasis becomes increasingly complex as more loci are considered due to the exponential expansion of possible interactions. Consequently, selecting key features that influence epistatic interactions is crucial for effective downstream analyses. Recognizing this challenge, this study investigates the efficiency of Relief-Based Algorithms (RBAs) in detecting higher-order epistatic interactions, which may be critical for understanding the genetic architecture of complex traits. RBAs are uniquely non-exhaustive, eliminating the need to construct features for every possible interaction and thus improving computational tractability. Motivated by previous research indicating that some RBAs rank predictive features involved in higher-order epistasis as highly negative, we explore the utility of absolute value ranking of RBA feature weights as an alternative method to capture complex interactions. We evaluate ReliefF, MultiSURF, and MultiSURFstar on simulated genetic datasets that model various patterns of genotype-phenotype associations, including 2-way to 5-way genetic interactions, and compare their performance to two control methods: a random shuffle and mutual information. Our findings indicate that while RBAs effectively identify lower-order (2 to 3-way) interactions, their capability to detect higher-order interactions is significantly limited, primarily by large feature count but also by signal noise. Specifically, we observe that RBAs are successful in detecting fully penetrant 4-way XOR interactions using an absolute value ranking approach, but this is restricted to datasets with a minimal number of total features. These results highlight the inherent limitations of current RBAs and underscore the need for enhanced detection capabilities for the investigation of epistasis, particularly in datasets with large feature counts and complex higher-order interactions.

Higher-order interactions and zero-determinant strategies in the public goods game

Abstract Since the ingenious discovery of zero-determinant (ZD) strategies by Press and Dyson, many efforts have been devoted to the evolutionary performance of ZD strategies. Recently, the effects of higher-order interactions on evolutionary games have attracted widespread interests, whereas it remains unknown how higher-order interactions affect the evolutionary performance of ZD strategies. This paper focuses on the role of higher-order interactions on evolutionary ZD strategies in iterated public goods game (IPGG), where the baseline payoff is a key parameter to describe nodes' extent of reciprocity in both first-order and second-order interactions. Through the adaptive-like dynamics, we found that there is a critical value of each network, above which the networked game will converge to a consensus state where all the nodes obtain the same payoff. This critical value is significantly affected by the relative strength of higher-order interactions with a U-shaped trend. Numerical simulations are carried out to explore how the network structures affect the dynamics. The results in networks with different sizes indicate that networks with higher average degree are more easily to converge to the consensus state. The simulations on a real-world network further support the theoretical conclusions.

Fractal information dissemination and clustering evolution on social hypernetwork

The complexity of systems stems from the richness of the group interactions among their units. Classical networks exhibit identified limits in the study of complex systems, where links connect pairs of nodes, inability to comprehensively describe higher-order interactions in networks. Higher-order networks can enhance modeling capacities of group interaction networks and help understand and predict network dynamical behavior. This paper constructs a social hypernetwork with a group structure by analyzing a community overlapping structure and a network iterative relationship, and the overlapping relationship between communities is logically separated. Considering the different group behavior pattern and attention focus, we defined the group cognitive disparity, group credibility, group cohesion index, hyperedge strength to study the relationship between information dissemination and network evolution. This study shows that groups can alter the connected network through information propagation, and users in social networks tend to form highly connected groups or communities in information dissemination. Propagation networks with high clustering coefficients promote the fractal information dissemination, which in itself drives the fractal evolution of groups within the network. This study emphasizes the significant role of “key groups” with overlapping structures among communities in group network propagation. Real cases provide evidence for the clustering phenomenon and fractal evolution of networks.

Source inference for misinformation spreading on hypergraphs

Source inference aims at revealing the seed of the misinformation spreading on social networks, and attracted great attention in the field of network science and cybersecurity. Extensive real-world data analyses have certificated that individual interactions exist pairwise and higher-order interactions, and thus should be described using the hypergraph. Previous studies about the source inference algorithms are mainly focused on simple graphs (i.e., a graph only has pairwise interactions) while neglecting the higher-order interactions. In this article, we propose a dynamical message-passing (DMP) algorithm to infer the misinformation spreading on hypergraphs. As a comparison, we also extend jordan centrality (JC), betweenness centrality (BC), closeness centrality (CC), and eigenvector centrality (EIG) algorithm to hypergraphs. The results show that our proposed DMP algorithm can accurately and effectively infer the propagation source on both artificial and real-world hypergraphs compared to the centrality algorithms. Even if the increase in propagation scale renders the other centrality algorithms completely ineffective, the DMP algorithm can still distinguish the real propagation sources in the form of a rank smaller than 10% of the network size. In general, our DMP algorithm provides an effective solution for inferring the misinformation source on high-order networks.

Can a Transparent Machine Learning Algorithm Predict Better than Its Black Box Counterparts? A Benchmarking Study Using 110 Data Sets.

We developed a novel machine learning (ML) algorithm with the goal of producing transparent models (i.e., understandable by humans) while also flexibly accounting for nonlinearity and interactions. Our method is based on ranked sparsity, and it allows for flexibility and user control in varying the shade of the opacity of black box machine learning methods. The main tenet of ranked sparsity is that an algorithm should be more skeptical of higher-order polynomials and interactions a priori compared to main effects, and hence, the inclusion of these more complex terms should require a higher level of evidence. In this work, we put our new ranked sparsity algorithm (as implemented in the open source R package, sparseR) to the test in a predictive model "bakeoff" (i.e., a benchmarking study of ML algorithms applied "out of the box", that is, with no special tuning). Algorithms were trained on a large set of simulated and real-world data sets from the Penn Machine Learning Benchmarks database, addressing both regression and binary classification problems. We evaluated the extent to which our human-centered algorithm can attain predictive accuracy that rivals popular black box approaches such as neural networks, random forests, and support vector machines, while also producing more interpretable models. Using out-of-bag error as a meta-outcome, we describe the properties of data sets in which human-centered approaches can perform as well as or better than black box approaches. We found that interpretable approaches predicted optimally or within 5% of the optimal method in most real-world data sets. We provide a more in-depth comparison of the performances of random forests to interpretable methods for several case studies, including exemplars in which algorithms performed similarly, and several cases when interpretable methods underperformed. This work provides a strong rationale for including human-centered transparent algorithms such as ours in predictive modeling applications.

Death transitions in attractive–repulsive coupled oscillators with higher-order interactions

The first-order phase transition from oscillation to steady state, known as explosive death (ED), is prevalent in various dynamical models. However, previous studies on ED have predominantly focused on interactions between pairs of elements. In this work, we investigate how death transitions occur when both pairwise and three-body interactions are present in an extended Van der Pol oscillator network with attractive–repulsive coupling. By examining both global and non-local interaction mechanisms, the impact of higher-order interactions on ED and the differences in the transition process are comprehensively analyzed. Firstly, we construct a diagram of the global dynamics in the context of higher-order and first-order coupling strengths, identifying that the higher-order interactions promote the onset of ED with a contribution comparable to that of first-order interactions. Specifically, for global coupling, the theoretical backward critical curves matching the numerical results are derived through linear stability analyses, showcasing a linear correlation with a slope of -1 between the higher-order and first-order coupling strengths. Under non-local coupling, fitting the numerically obtained backward critical curves likewise yields a consistent quantitative relationship. Additionally, during the transition process of ED, we discover intriguing coexisting states in the hysteresis area under non-local coupling, including the coexistence of chimera states with coherent or incoherent oscillation, and the coexistence of chimera states with oscillation death. This is attributed to symmetry breaking induced by non-local action. These findings enhance the understanding of higher-order interactions in complex systems and provide a fresh perspective for studying multi-stability behavior in biochemical and physical systems.

Opinion dynamics in social networks incorporating higher-order interactions

Opinion dynamics in social networks incorporating higher-order interactions

A high-order focus interaction model and oral ulcer dataset for oral ulcer segmentation

Computer-aided diagnosis has been slow to develop in the field of oral ulcers. One of the major reasons for this is the lack of publicly available datasets. However, oral ulcers have cancerous lesions and their mortality rate is high. The ability to recognize oral ulcers at an early stage in a timely and effective manner is a very critical issue. In recent years, although there exists a small group of researchers working on these, the datasets are private. Therefore to address this challenge, in this paper a multi-tasking oral ulcer dataset (Autooral) containing two major tasks of lesion segmentation and classification is proposed and made publicly available. To the best of our knowledge, we are the first team to make publicly available an oral ulcer dataset with multi-tasking. In addition, we propose a novel modeling framework, HF-UNet, for segmenting oral ulcer lesion regions. Specifically, the proposed high-order focus interaction module (HFblock) performs acquisition of global properties and focus for acquisition of local properties through high-order attention. The proposed lesion localization module (LL-M) employs a novel hybrid sobel filter, which improves the recognition of ulcer edges. Experimental results on the proposed Autooral dataset show that our proposed HF-UNet segmentation of oral ulcers achieves a DSC value of about 0.80 and the inference memory occupies only 2029 MB. The proposed method guarantees a low running load while maintaining a high-performance segmentation capability. The proposed Autooral dataset and code are available from https://github.com/wurenkai/HF-UNet-and-Autooral-dataset.

Modelling network motifs as higher order interactions: a statistical inference based approach

The prevalent approach to motif analysis seeks to describe the local connectivity structure of networks by identifying subgraph patterns that appear significantly more often in a network then expected under a null model that conserves certain features of the original network. In this article we advocate for an alternative approach based on statistical inference of generative models where nodes are connected not only by edges but also copies of higher order subgraphs. These models naturally lead to the consideration of latent states that correspond to decompositions of networks into higher order interactions in the form of subgraphs that can have the topology of any simply connected motif. Being based on principles of parsimony the method can infer concise sets of motifs from within thousands of candidates allowing for consistent detection of larger motifs. The inferential approach yields not only a set of statistically significant higher order motifs but also an explicit decomposition of the network into these motifs, which opens new possibilities for the systematic study of the topological and dynamical implications of higher order connectivity structures in networks. After briefly reviewing core concepts and methods, we provide example applications to empirical data sets and discuss how the inferential approach addresses current problems in motif analysis and explore how concepts and methods common to motif analysis translate to the inferential framework.

Structure of higher-order interactions in social-ecological networks through Q-analysis of their neighbourhood and clique complex.

This paper studies higher-order interactions in social-ecological networks, which formally represent interactions within the social and ecological units of an ecosystem. Many real-world social ecosystems exhibit not only pairwise interactions but also higher-order interactions among their units. Therefore, the conventional graph-theoretic description of networks falls short of capturing these higher-order interactions due to the inherent limitations of the graph definition. In this work, a mathematical framework for capturing the higher-order interactions of a social-ecological system has been given by incorporating notions from combinatorial algebraic topology. In order to achieve this, two different simplicial complexes, the clique and the neighbourhood complex, have been constructed from a pairwise social-ecological network. As a case study, the Q-analysis and a structural study of the interactions in the rural agricultural system of southern Madagascar have been done at various structural levels denoted by q. The results obtained by calculating all the structural vectors for both simplicial complexes, along with exciting results about the participation of facets of the clique complex at different q-levels, have been discussed. This work also establishes significant theorems concerning the dimension of the neighbourhood complex and clique complex obtained from the parent pairwise network.

Brain clocks capture diversity and disparities in aging and dementia across geographically diverse populations.

Brain clocks, which quantify discrepancies between brain age and chronological age, hold promise for understanding brain health and disease. However, the impact of diversity (including geographical, socioeconomic, sociodemographic, sex and neurodegeneration) on the brain-age gap is unknown. We analyzed datasets from 5,306 participants across 15 countries (7 Latin American and Caribbean countries (LAC) and 8 non-LAC countries). Based on higher-order interactions, we developed a brain-age gap deep learning architecture for functional magnetic resonance imaging (2,953) and electroencephalography (2,353). The datasets comprised healthy controls and individuals with mild cognitive impairment, Alzheimer disease and behavioral variant frontotemporal dementia. LAC models evidenced older brain ages (functional magnetic resonance imaging: mean directional error = 5.60, root mean square error (r.m.s.e.) = 11.91; electroencephalography: mean directional error = 5.34, r.m.s.e. = 9.82) associated with frontoposterior networks compared with non-LAC models. Structural socioeconomic inequality, pollution and health disparities were influential predictors of increased brain-age gaps, especially in LAC (R² = 0.37, F² = 0.59, r.m.s.e. = 6.9). An ascending brain-age gap from healthy controls to mild cognitive impairment to Alzheimer disease was found. In LAC, we observed larger brain-age gaps in females in control and Alzheimer disease groups compared with the respective males. The results were not explained by variations in signal quality, demographics or acquisition methods. These findings provide a quantitative framework capturing the diversity of accelerated brain aging.