From networks to optimal higher-order models of complex systems

Counterfactual learning for higher-order relation prediction in heterogeneous information networks

Heterogeneous Information Networks (HINs) play a crucial role in modeling complex social systems, where predicting missing links/relations is a significant task. Existing methods primarily focus on pairwise relations, but real-world scenarios often involve multi-entity interactions. For example, in academic collaboration networks, an interaction occurs between a paper, a conference, and multiple authors. These higher-order relations are prevalent but have been underexplored. Moreover, existing methods often neglect the causal relationship between the global graph structure and the state of relations, limiting their ability to capture the fundamental factors driving relation prediction. In this paper, we propose HINCHOR, an end-to-end model for higher-order relation prediction in HINs. HINCHOR introduces a higher-order structure encoder to capture multi-entity proximity information. Then, it focuses on a counterfactual question: "If the global graph structure were different, would the higher-order relation change?" By presenting a counterfactual data augmentation module, HINCHOR utilizes global structure information to generate counterfactual relations. Through counterfactual learning, HINCHOR estimates causal effects while predicting higher-order relations. The experimental results on four constructed benchmark datasets show that HINCHOR outperforms existing state-of-the-art methods.

Cascading failure dynamics on higher-order networks with load redistribution

The phenomenon of load redistribution in complex networks has garnered extensive attention due to its profound impact and widespread occurrence. In recent years, higher-order structures have offered new insights into understanding the structures and dynamic processes of complex networks. However, the influence of these higher-order structures on the dynamics of load redistribution, cascade failures, and recovery processes remains to be fully explored. In this study, we propose the load redistribution model with higher-order structures and recovery strategies of cascade failure based on functional upgrading and reconstruction mechanisms. In the cascading failure process with load redistribution and higher-order recovery strategies, we find that higher-order structures can induce a discontinuous phase transition at the low proportion of load redistribution, and the dynamic process displays a dual character of being robust yet fragile. These findings have been examined in both real and classical modeled networks. Interestingly, the largest connected component exhibits three distinct modes as the attack ratio increases at high densities of higher-order structures and recovery mechanisms.

Detailed-level modelling of influence spreading on complex networks

The progress in high-performance computing makes it increasingly possible to build detailed models to investigate spreading processes on complex networks. However, current studies have been lacking detailed computational methods to describe spreading processes in large complex networks. To fill this gap we present a new modelling approach for analysing influence spreading via individual nodes and links on various network structures. The proposed influence-spreading model uses a probability matrix to capture the spreading probability from one node to another in the network. This approach enables analysing network characteristics in a number of applications and spreading processes using metrics that are consistent with the quantities used to model the network structures. In addition, this study combines sub-models and offers a comprehensive look at different applications and metrics previously discussed in cases of social networks, community detection, and epidemic spreading. Here, we also note that the centrality measures based on the probability matrix are used to identify the most significant nodes in the network. Furthermore, the model can be expanded to include additional properties, such as introducing individual breakthrough probabilities for the nodes and specific temporal distributions for the links.

Complex system modelling reveals oxalate homeostasis is driven by diverse oxalate-degrading bacteria

Decades of research have made clear that host-associated microbiomes touch all facets of health. However, effective therapies that target the microbiome have been elusive given its inherent complexity. Here, we experimentally examined diet-microbe-host interactions through a complex systems framework, centered on dietary oxalate. Using multiple, independent molecular, animal, and in vitro experimental models, we found that microbiome composition influenced multiple oxalate-microbe-host interfaces. Importantly, administration of the oxalate-degrading specialist, Oxalobacter formigenes, was only effective against a poor oxalate-degrading microbiota background and gives critical new insights into why clinical intervention trials with this species exhibit variable outcomes. Data suggest that, while heterogeneity in the microbiome impacts multiple diet-host-microbe interfaces, metabolic redundancy among diverse microorganisms in specific diet-microbe axes is a critical variable that may impact the efficacy of bacteriotherapies, which can help guide patient and probiotic selection criteria in probiotic clinical trials.

Deeper but smaller: Higher-order interactions increase linear stability but shrink basins

A key challenge of nonlinear dynamics and network science is to understand how higher-order interactions influence collective dynamics. Although many studies have approached this question through linear stability analysis, less is known about how higher-order interactions shape the global organization of different states. Here, we shed light on this issue by analyzing the rich patterns supported by identical Kuramoto oscillators on hypergraphs. We show that higher-order interactions can have opposite effects on linear stability and basin stability: They stabilize twisted states (including full synchrony) by improving their linear stability, but also make them hard to find by markedly reducing their basin size. Our results highlight the importance of understanding higher-order interactions from both local and global perspectives.

Hypernetwork modeling and topology of high-order interactions for complex systems

Interactions among the underlying agents of a complex system are not only limited to dyads but can also occur in larger groups. Currently, no generic model has been developed to capture high-order interactions (HOI), which, along with pairwise interactions, portray a detailed landscape of complex systems. Here, we integrate evolutionary game theory and behavioral ecology into a unified statistical mechanics framework, allowing all agents (modeled as nodes) and their bidirectional, signed, and weighted interactions at various orders (modeled as links or hyperlinks) to be coded into hypernetworks. Such hypernetworks can distinguish between how pairwise interactions modulate a third agent (active HOI) and how the altered state of each agent in turn governs interactions between other agents (passive HOI). The simultaneous occurrence of active and passive HOI can drive complex systems to evolve at multiple time and space scales. We apply the model to reconstruct a hypernetwork of hexa-species microbial communities, and by dissecting the topological architecture of the hypernetwork using GLMY homology theory, we find distinct roles of pairwise interactions and HOI in shaping community behavior and dynamics. The statistical relevance of the hypernetwork model is validated using a series of in vitro mono-, co-, and tricultural experiments based on three bacterial species.

Ambiguities in neural-network-based hyperedge prediction

A hypergraph is a generalization of a graph that depicts higher-order relations. Predicting higher-order relations, i.e. hyperedges, is a fundamental problem in hypergraph studies, and has immense applications in multiple domains. Recent development of graph neural network (GNN) advanced the prediction of pair-wise relations in graphs. However, existing methods can hardly be extended to hypergraphs due to the lack of higher-order dependency in their graph embedding. In this paper, we mathematically formulate the ambiguity challenges of GNN-based representation of higher-order relations, namely node-level and hyperedge-level ambiguities. We further present HIGNN (Hyperedge Isomorphism Graph Neural Network) that utilizes bipartite graph neural network with hyperedge structural features to collectively tackle the two ambiguity issues in the hyperedge prediction problem. HIGNN achieves constant performance improvement compared with recent GNN-based models. In addition, we apply HIGNN to a new task, predicting genetic higher-order interactions on 3D genome organization data. HIGNN shows consistently higher prediction accuracy across different chromosomes, and generates novel findings on 4-way gene interactions, which is further validated by existing literature.

A simplex path integral and a simplex renormalization group for high-order interactions. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2024;87:087601. [PMID: 39077989 DOI: 10.1088/1361-6633/ad5c99] [Citation(s) in RCA: 0] [Impact Index Per Article: 0]

Modern theories of phase transitions and scale invariance are rooted in path integral formulation and renormalization groups (RGs). Despite the applicability of these approaches in simple systems with only pairwise interactions, they are less effective in complex systems with undecomposable high-order interactions (i.e. interactions among arbitrary sets of units). To precisely characterize the universality of high-order interacting systems, we propose a simplex path integral and a simplex RG (SRG) as the generalizations of classic approaches to arbitrary high-order and heterogeneous interactions. We first formalize the trajectories of units governed by high-order interactions to define path integrals on corresponding simplices based on a high-order propagator. Then, we develop a method to integrate out short-range high-order interactions in the momentum space, accompanied by a coarse graining procedure functioning on the simplex structure generated by high-order interactions. The proposed SRG, equipped with a divide-and-conquer framework, can deal with the absence of ergodicity arising from the sparse distribution of high-order interactions and can renormalize a system with intertwined high-order interactions at thep-order according to its properties at theq-order (p⩽q). The associated scaling relation and its corollaries provide support to differentiate among scale-invariant, weakly scale-invariant, and scale-dependent systems across different orders. We validate our theory in multi-order scale-invariance verification, topological invariance discovery, organizational structure identification, and information bottleneck analysis. These experiments demonstrate the capability of our theory to identify intrinsic statistical and topological properties of high-order interacting systems during system reduction.

Shortest-Path Percolation on Random Networks

We propose a bond-percolation model intended to describe the consumption, and eventual exhaustion, of resources in transport networks. Edges forming minimum-length paths connecting demanded origin-destination nodes are removed if below a certain budget. As pairs of nodes are demanded and edges are removed, the macroscopic connected component of the graph disappears, i.e., the graph undergoes a percolation transition. Here, we study such a shortest-path-percolation transition in homogeneous random graphs where pairs of demanded origin-destination nodes are randomly generated, and fully characterize it by means of finite-size scaling analysis. If budget is finite, the transition is identical to the one of ordinary percolation, where a single giant cluster shrinks as edges are removed from the graph; for infinite budget, the transition becomes more abrupt than the one of ordinary percolation, being characterized by the sudden fragmentation of the giant connected component into a multitude of clusters of similar size.

Learning interpretable dynamics of stochastic complex systems from experimental data

Complex systems with many interacting nodes are inherently stochastic and best described by stochastic differential equations. Despite increasing observation data, inferring these equations from empirical data remains challenging. Here, we propose the Langevin graph network approach to learn the hidden stochastic differential equations of complex networked systems, outperforming five state-of-the-art methods. We apply our approach to two real systems: bird flock movement and tau pathology diffusion in brains. The inferred equation for bird flocks closely resembles the second-order Vicsek model, providing unprecedented evidence that the Vicsek model captures genuine flocking dynamics. Moreover, our approach uncovers the governing equation for the spread of abnormal tau proteins in mouse brains, enabling early prediction of tau occupation in each brain region and revealing distinct pathology dynamics in mutant mice. By learning interpretable stochastic dynamics of complex systems, our findings open new avenues for downstream applications such as control.

The neuron mixer and its impact on human brain dynamics

A signal mixer facilitates rich computation, which has been the building block of modern telecommunication. This frequency mixing produces new signals at the sum and difference frequencies of input signals, enabling powerful operations such as heterodyning and multiplexing. Here, we report that a neuron is a signal mixer. We found through ex vivo and in vivo whole-cell measurements that neurons mix exogenous (controlled) and endogenous (spontaneous) subthreshold membrane potential oscillations, producing new oscillation frequencies, and that neural mixing originates in voltage-gated ion channels. Furthermore, we demonstrate that mixing is evident in human brain activity and is associated with cognitive functions. We found that the human electroencephalogram displays distinct clusters of local and inter-region mixing and that conversion of the salient posterior alpha-beta oscillations into gamma-band oscillations regulates visual attention. Signal mixing may enable individual neurons to sculpt the spectrum of neural circuit oscillations and utilize them for computational operations.

From unbiased to maximal-entropy random walks on hypergraphs

Random walks have been intensively studied on regular and complex networks, which are used to represent pairwise interactions. Nonetheless, recent works have demonstrated that many real-world processes are better captured by higher-order relationships, which are naturally represented by hypergraphs. Here we study random walks on hypergraphs. Due to the higher-order nature of these mathematical objects, one can define more than one type of walks. In particular, we study the unbiased and the maximal entropy random walk on hypergraphs with two types of steps, emphasizing their similarities and differences. We characterize these dynamic processes by examining their stationary distributions and associated hitting times. To illustrate our findings, we present a toy example and conduct extensive analyses of artificial and real hypergraphs, providing insights into both their structural and dynamical properties. We hope that our findings motivate further research extending the analysis to different classes of random walks as well as to practical applications.

Legal hypergraphs

Complexity science provides a powerful framework for understanding physical, biological and social systems, and network analysis is one of its principal tools. Since many complex systems exhibit multilateral interactions that change over time, in recent years, network scientists have become increasingly interested in modelling and measuring dynamic networks featuring higher-order relations. At the same time, while network analysis has been more widely adopted to investigate the structure and evolution of law as a complex system, the utility of dynamic higher-order networks in the legal domain has remained largely unexplored. Setting out to change this, we introduce temporal hypergraphs as a powerful tool for studying legal network data. Temporal hypergraphs generalize static graphs by (i) allowing any number of nodes to participate in an edge and (ii) permitting nodes or edges to be added, modified or deleted. We describe models and methods to explore legal hypergraphs that evolve over time and elucidate their benefits through case studies on legal citation and collaboration networks that change over a period of more than 70 years. Our work demonstrates the potential of dynamic higher-order networks for studying complex legal systems, and it facilitates further advances in legal network analysis. This article is part of the theme issue 'A complexity science approach to law and governance'.

Does the brain behave like a (complex) network? I. Dynamics

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.

Fundamental patterns of signal propagation in complex networks

Various disasters stem from minor perturbations, such as the spread of infectious diseases and cascading failure in power grids. Analyzing perturbations is crucial for both theoretical and application fields. Previous researchers have proposed basic propagation patterns for perturbation and explored the impact of basic network motifs on the collective response to these perturbations. However, the current framework is limited in its ability to decouple interactions and, therefore, cannot analyze more complex structures. In this article, we establish an effective, robust, and powerful propagation framework under a general dynamic model. This framework reveals classical and dense network motifs that exert critical acceleration on signal propagation, often reducing orders of magnitude compared with conclusions generated by previous work. Moreover, our framework provides a new approach to understand the fundamental principles of complex systems and the negative feedback mechanism, which is of great significance for researching system controlling and network resilience.

Evidence for increased parallel information transmission in human brain networks compared to macaques and male mice

Brain communication, defined as information transmission through white-matter connections, is at the foundation of the brain's computational capacities that subtend almost all aspects of behavior: from sensory perception shared across mammalian species, to complex cognitive functions in humans. How did communication strategies in macroscale brain networks adapt across evolution to accomplish increasingly complex functions? By applying a graph- and information-theory approach to assess information-related pathways in male mouse, macaque and human brains, we show a brain communication gap between selective information transmission in non-human mammals, where brain regions share information through single polysynaptic pathways, and parallel information transmission in humans, where regions share information through multiple parallel pathways. In humans, parallel transmission acts as a major connector between unimodal and transmodal systems. The layout of information-related pathways is unique to individuals across different mammalian species, pointing at the individual-level specificity of information routing architecture. Our work provides evidence that different communication patterns are tied to the evolution of mammalian brain networks.

What Is in a Simplicial Complex? A Metaplex-Based Approach to Its Structure and Dynamics

Geometric realization of simplicial complexes makes them a unique representation of complex systems. The existence of local continuous spaces at the simplices level with global discrete connectivity between simplices makes the analysis of dynamical systems on simplicial complexes a challenging problem. In this work, we provide some examples of complex systems in which this representation would be a more appropriate model of real-world phenomena. Here, we generalize the concept of metaplexes to embrace that of geometric simplicial complexes, which also includes the definition of dynamical systems on them. A metaplex is formed by regions of a continuous space of any dimension interconnected by sinks and sources that works controlled by discrete (graph) operators. The definition of simplicial metaplexes given here allows the description of the diffusion dynamics of this system in a way that solves the existing problems with previous models. We make a detailed analysis of the generalities and possible extensions of this model beyond simplicial complexes, e.g., from polytopal and cell complexes to manifold complexes, and apply it to a real-world simplicial complex representing the visual cortex of a macaque.

Percolation Theories for Quantum Networks

Quantum networks have experienced rapid advancements in both theoretical and experimental domains over the last decade, making it increasingly important to understand their large-scale features from the viewpoint of statistical physics. This review paper discusses a fundamental question: how can entanglement be effectively and indirectly (e.g., through intermediate nodes) distributed between distant nodes in an imperfect quantum network, where the connections are only partially entangled and subject to quantum noise? We survey recent studies addressing this issue by drawing exact or approximate mappings to percolation theory, a branch of statistical physics centered on network connectivity. Notably, we show that the classical percolation frameworks do not uniquely define the network's indirect connectivity. This realization leads to the emergence of an alternative theory called "concurrence percolation", which uncovers a previously unrecognized quantum advantage that emerges at large scales, suggesting that quantum networks are more resilient than initially assumed within classical percolation contexts, offering refreshing insights into future quantum network design.

Constructing temporal networks with bursty activity patterns

Human social interactions tend to vary in intensity over time, whether they are in person or online. Variable rates of interaction in structured populations can be described by networks with the time-varying activity of links and nodes. One of the key statistics to summarize temporal patterns is the inter-event time, namely the duration between successive pairwise interactions. Empirical studies have found inter-event time distributions that are heavy-tailed, for both physical and digital interactions. But it is difficult to construct theoretical models of time-varying activity on a network that reproduce the burstiness seen in empirical data. Here we develop a spanning-tree method to construct temporal networks and activity patterns with bursty behavior. Our method ensures any desired target inter-event time distributions for individual nodes and links, provided the distributions fulfill a consistency condition, regardless of whether the underlying topology is static or time-varying. We show that this model can reproduce burstiness found in empirical datasets, and so it may serve as a basis for studying dynamic processes in real-world bursty interactions.

The metabolomic physics of complex diseases

Human diseases involve metabolic alterations. Metabolomic profiles have served as a vital biomarker for the early identification of high-risk individuals and disease prevention. However, current approaches can only characterize individual key metabolites, without taking into account the reality that complex diseases are multifactorial, dynamic, heterogeneous, and interdependent. Here, we leverage a statistical physics model to combine all metabolites into bidirectional, signed, and weighted interaction networks and trace how the flow of information from one metabolite to the next causes changes in health state. Viewing a disease outcome as the consequence of complex interactions among its interconnected components (metabolites), we integrate concepts from ecosystem theory and evolutionary game theory to model how the health state-dependent alteration of a metabolite is shaped by its intrinsic properties and through extrinsic influences from its conspecifics. We code intrinsic contributions as nodes and extrinsic contributions as edges into quantitative networks and implement GLMY homology theory to analyze and interpret the topological change of health state from symbiosis to dysbiosis and vice versa. The application of this model to real data allows us to identify several hub metabolites and their interaction webs, which play a part in the formation of inflammatory bowel diseases. The findings by our model could provide important information on drug design to treat these diseases and beyond.

Bayesian inference of transition matrices from incomplete graph data with a topological prior

Many network analysis and graph learning techniques are based on discrete- or continuous-time models of random walks. To apply these methods, it is necessary to infer transition matrices that formalize the underlying stochastic process in an observed graph. For weighted graphs, where weighted edges capture observations of repeated interactions between nodes, it is common to estimate the entries of such transition matrices based on the (relative) weights of edges. However in real-world settings we are often confronted with incomplete data, which turns the construction of the transition matrix based on a weighted graph into an inference problem. Moreover, we often have access to additional information, which capture topological constraints of the system, i.e. which edges in a weighted graph are (theoretically) possible and which are not. Examples include transportation networks, where we may have access to a small sample of passenger trajectories as well as the physical topology of connections, or a limited set of observed social interactions with additional information on the underlying social structure. Combining these two different sources of information to reliably infer transition matrices from incomplete data on repeated interactions is an important open challenge, with severe implications for the reliability of downstream network analysis tasks. Addressing this issue, we show that including knowledge on such topological constraints can considerably improve the inference of transition matrices, especially in situations where we only have a small number of observed interactions. To this end, we derive an analytically tractable Bayesian method that uses repeated interactions and a topological prior to perform data-efficient inference of transition matrices. We compare our approach against commonly used frequentist and Bayesian approaches both in synthetic data and in five real-world datasets, and we find that our method recovers the transition probabilities with higher accuracy. Furthermore, we demonstrate that the method is robust even in cases when the knowledge of the topological constraint is partial. Lastly, we show that this higher accuracy improves the results for downstream network analysis tasks like cluster detection and node ranking, which highlights the practical relevance of our method for interdisciplinary data-driven analyses of networked systems.

Locating community smells in software development processes using higher-order network centralities

Community smells are negative patterns in software development teams' interactions that impede their ability to successfully create software. Examples are team members working in isolation, lack of communication and collaboration across departments or sub-teams, or areas of the codebase where only a few team members can work on. Current approaches aim to detect community smells by analysing static network representations of software teams' interaction structures. In doing so, they are insufficient to locate community smells within development processes. Extending beyond the capabilities of traditional social network analysis, we show that higher-order network models provide a robust means of revealing such hidden patterns and complex relationships. To this end, we develop a set of centrality measures based on the MOGen higher-order network model and show their effectiveness in predicting influential nodes using five empirical datasets. We then employ these measures for a comprehensive analysis of a product team at the German IT security company genua GmbH, showcasing our method's success in identifying and locating community smells. Specifically, we uncover critical community smells in two areas of the team's development process. Semi-structured interviews with five team members validate our findings: while the team was aware of one community smell and employed measures to address it, it was not aware of the second. This highlights the potential of our approach as a robust tool for identifying and addressing community smells in software development teams. More generally, our work contributes to the social network analysis field with a powerful set of higher-order network centralities that effectively capture community dynamics and indirect relationships.

Hyper-cores promote localization and efficient seeding in higher-order processes

Going beyond networks, to include higher-order interactions of arbitrary sizes, is a major step to better describe complex systems. In the resulting hypergraph representation, tools to identify structures and central nodes are scarce. We consider the decomposition of a hypergraph in hyper-cores, subsets of nodes connected by at least a certain number of hyperedges of at least a certain size. We show that this provides a fingerprint for data described by hypergraphs and suggests a novel notion of centrality, the hypercoreness. We assess the role of hyper-cores and nodes with large hypercoreness in higher-order dynamical processes: such nodes have large spreading power and spreading processes are localized in central hyper-cores. Additionally, in the emergence of social conventions very few committed individuals with high hypercoreness can rapidly overturn a majority convention. Our work opens multiple research avenues, from comparing empirical data to model validation and study of temporally varying hypergraphs.

The characteristics and influencing factors of spatial network of city-based innovation correlation in China: from the perspective of high tech zones

In the context of "space of flows", city-based innovation correlation in driving economic growth is no longer limited to the traditional hierarchical structure. It is of great significance to explore Chinese cities innovation association network from the perspective of high-tech zones which gather a large number of innovation resources. Here our report is to provide new ideas for improving the innovation capability of high-tech zones and accelerating the construction of Chinese high-quality innovation system. Here we take 142 cities with high-tech zones as research samples, and explore the characteristics and influencing factors of spatial network of city-based innovation correlation in China, through modified gravity modelsocial, network analysis and QAP analysis. The results show that city-based innovation network is not closely connected, the number of redundant connection channels is low efficiency, showing a four-level spatial pattern of "Z" shaped spindle. Among them, degree centrality of cities in eastern China is higher than that in the western region, the core cities in central China play a bridging role, and western remote cities are easily affected by related cities. Moreover, there are four innovation cohesion subgroups, including the northern hinterland subgroup, the eastern coastal subgroup, the southern subgroup and the western cooperation subgroup. Furthermore, the results of the influencing factors analysis show the differences in administrative level, economic development level, openness to the outside world, and investment in technology are conducive to the innovation association between cities, while the similarities in spatial adjacency and industrial structure will promote the strong innovation association between cities.

Predicting variable-length paths in networked systems using multi-order generative models

Apart from nodes and links, for many networked systems, we have access to data on paths, i.e., collections of temporally ordered variable-length node sequences that are constrained by the system's topology. Understanding the patterns in such data is key to advancing our understanding of the structure and dynamics of complex systems. Moreover, the ability to accurately model and predict paths is important for engineered systems, e.g., to optimise supply chains or provide smart mobility services. Here, we introduce MOGen, a generative modelling framework that enables both next-element and out-of-sample prediction in paths with high accuracy and consistency. It features a model selection approach that automatically determines the optimal model directly from data, effectively making MOGen parameter-free. Using empirical data, we show that our method outperforms state-of-the-art sequence modelling techniques. We further introduce a mathematical formalism that links higher-order models of paths to transition matrices of random walks in multi-layer networks.

Inference of time-ordered multibody interactions

We introduce time-ordered multibody interactions to describe complex systems manifesting temporal as well as multibody dependencies. First, we show how the dynamics of multivariate Markov chains can be decomposed in ensembles of time-ordered multibody interactions. Then, we present an algorithm to extract those interactions from data capturing the system-level dynamics of node states and a measure to characterize the complexity of interaction ensembles. Finally, we experimentally validate the robustness of our algorithm against statistical errors and its efficiency at inferring parsimonious interaction ensembles.

Fixation dynamics on hypergraphs

Hypergraphs have been a useful tool for analyzing population dynamics such as opinion formation and the public goods game occurring in overlapping groups of individuals. In the present study, we propose and analyze evolutionary dynamics on hypergraphs, in which each node takes one of the two types of different but constant fitness values. For the corresponding dynamics on conventional networks, under the birth-death process and uniform initial conditions, most networks are known to be amplifiers of natural selection; amplifiers by definition enhance the difference in the strength of the two competing types in terms of the probability that the mutant type fixates in the population. In contrast, we provide strong computational evidence that a majority of hypergraphs are suppressors of selection under the same conditions by combining theoretical and numerical analyses. We also show that this suppressing effect is not explained by one-mode projection, which is a standard method for expressing hypergraph data as a conventional network. Our results suggest that the modeling framework for structured populations in addition to the specific network structure is an important determinant of evolutionary dynamics, paving a way to studying fixation dynamics on higher-order networks including hypergraphs.

Distributed Consensus Algorithms in Sensor Networks with Higher-Order Topology

Information aggregation in distributed sensor networks has received significant attention from researchers in various disciplines. Distributed consensus algorithms are broadly developed to accelerate the convergence to consensus under different communication and/or energy limitations. Non-Bayesian social learning strategies are representative algorithms for distributed agents to learn progressively an underlying state of nature by information communications and evolutions. This work designs a new non-Bayesian social learning strategy named the hypergraph social learning by introducing the higher-order topology as the underlying communication network structure, with its convergence as well as the convergence rate theoretically analyzed. Extensive numerical examples are provided to demonstrate the effectiveness of the framework and reveal its superior performance when applying to sensor networks in tasks such as cooperative positioning. The designed framework can assist sensor network designers to develop more efficient communication topology, which can better resist environmental obstructions, and also has theoretical and applied values in broad areas such as distributed parameter estimation, dispersed information aggregation and social networks.

From single-neuron dynamics to higher-order circuit motifs in control and pathological brain networks

The convergence of advanced single-cell in vivo functional imaging techniques, computational modelling tools and graph-based network analytics has heralded new opportunities to study single-cell dynamics across large-scale networks, providing novel insights into principles of brain communication and pointing towards potential new strategies for treating neurological disorders. A major recent finding has been the identification of unusually richly connected hub cells that have capacity to synchronize networks and may also be critical in network dysfunction. While hub neurons are traditionally defined by measures that consider solely the number and strength of connections, novel higher-order graph analytics now enables the mining of massive networks for repeating subgraph patterns called motifs. As an illustration of the power offered by higher-order analysis of neuronal networks, we highlight how recent methodological advances uncovered a new functional cell type, the superhub, that is predicted to play a major role in regulating network dynamics. Finally, we discuss open questions that will be critical for assessing the importance of higher-order cellular-scale network analytics in understanding brain function in health and disease.

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.

Mitochondrial networks through the lens of mathematics

Mitochondria serve a wide range of functions within cells, most notably via their production of ATP. Although their morphology is commonly described as bean-like, mitochondria often form interconnected networks within cells that exhibit dynamic restructuring through a variety of physical changes. Further, though relationships between form and function in biology are well established, the extant toolkit for understanding mitochondrial morphology is limited. Here, we emphasize new and established methods for quantitatively describing mitochondrial networks, ranging from unweighted graph-theoretic representations to multi-scale approaches from applied topology, in particular persistent homology. We also show fundamental relationships between mitochondrial networks, mathematics, and physics, using ideas of graph planarity and statistical mechanics to better understand the full possible morphological space of mitochondrial network structures. Lastly, we provide suggestions for how examination of mitochondrial network form through the language of mathematics can inform biological understanding, and vice versa.

Identifying Key Factors of Hazardous Materials Transportation Accidents Based on Higher-Order and Multilayer Networks

This paper focuses on the application of higher-order and multilayer networks in identifying critical causes and relationships contributing to hazardous materials transportation accidents. There were 792 accidents of hazardous materials transportation that occurred on the road from 2017 to 2021 which have been investigated. By considering time sequence and dependency of causes, the hazardous materials transportation accidents causation network (HMTACN) was described using the higher-order model. To investigate the structure of HMTACN such as the importance of causes and links, HMTACN was divided into three layers using the weighted k-core decomposition: the core layer, the bridge layer and the peripheral layer. Then causes and links were analyzed in detail. It was found that the core layer was tightly connected and supported most of the causal flows of HMTACN. The results showed that causes should be given hierarchical attention. This study provides an innovative method to analyze complicated accidents, which can be used in identifying major causes and links. And this paper brings new ideas about safety network study and extends the applications of complex network theory.

Digital contact tracing on hypergraphs

The higher-order interactions emerging in the network topology affect the effectiveness of digital contact tracing (DCT). In this paper, we propose a mathematical model in which we use the hypergraph to describe the gathering events. In our model, the role of DCT is modeled as individuals carrying the app. When the individuals in the hyperedge all carry the app, epidemics cannot spread through this hyperedge. We develop a generalized percolation theory to investigate the epidemic outbreak size and threshold. We find that DCT can effectively suppress the epidemic spreading, i.e., decreasing the outbreak size and enlarging the threshold. DCT limits the spread of the epidemic to larger cardinality of hyperedges. On real-world networks, the inhibitory effect of DCT on the spread of epidemics is evident when the spread of epidemics is small.

Structure and function in artificial, zebrafish and human neural networks

Network science provides a set of tools for the characterization of the structure and functional behavior of complex systems. Yet a major problem is to quantify how the structural domain is related to the dynamical one. In other words, how the diversity of dynamical states of a system can be predicted from the static network structure? Or the reverse problem: starting from a set of signals derived from experimental recordings, how can one discover the network connections or the causal relations behind the observed dynamics? Despite the advances achieved over the last two decades, many challenges remain concerning the study of the structure-dynamics interplay of complex systems. In neuroscience, progress is typically constrained by the low spatio-temporal resolution of experiments and by the lack of a universal inferring framework for empirical systems. To address these issues, applications of network science and artificial intelligence to neural data have been rapidly growing. In this article, we review important recent applications of methods from those fields to the study of the interplay between structure and functional dynamics of human and zebrafish brain. We cover the selection of topological features for the characterization of brain networks, inference of functional connections, dynamical modeling, and close with applications to both the human and zebrafish brain. This review is intended to neuroscientists who want to become acquainted with techniques from network science, as well as to researchers from the latter field who are interested in exploring novel application scenarios in neuroscience.

Circulation of a digital community currency

Circulation is the characteristic feature of successful currency systems, from community currencies to cryptocurrencies to national currencies. In this paper, we propose a network analysis approach especially suited for studying circulation given a system's digital transaction records. Sarafu is a digital community currency that was active in Kenya over a period that saw considerable economic disruption due to the COVID-19 pandemic. We represent its circulation as a network of monetary flow among the 40,000 Sarafu users. Network flow analysis reveals that circulation was highly modular, geographically localized, and occurring among users with diverse livelihoods. Across localized sub-populations, network cycle analysis supports the intuitive notion that circulation requires cycles. Moreover, the sub-networks underlying circulation are consistently degree disassortative and we find evidence of preferential attachment. Community-based institutions often take on the role of local hubs, and network centrality measures confirm the importance of early adopters and of women's participation. This work demonstrates that networks of monetary flow enable the study of circulation within currency systems at a striking level of detail, and our findings can be used to inform the development of community currencies in marginalized areas.

Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes

Higher-order networks have emerged as a powerful framework to model complex systems and their collective behavior. Going beyond pairwise interactions, they encode structured relations among arbitrary numbers of units through representations such as simplicial complexes and hypergraphs. So far, the choice between simplicial complexes and hypergraphs has often been motivated by technical convenience. Here, using synchronization as an example, we demonstrate that the effects of higher-order interactions are highly representation-dependent. In particular, higher-order interactions typically enhance synchronization in hypergraphs but have the opposite effect in simplicial complexes. We provide theoretical insight by linking the synchronizability of different hypergraph structures to (generalized) degree heterogeneity and cross-order degree correlation, which in turn influence a wide range of dynamical processes from contagion to diffusion. Our findings reveal the hidden impact of higher-order representations on collective dynamics, highlighting the importance of choosing appropriate representations when studying systems with nonpairwise interactions.

Generative hypergraph models and spectral embedding

Many complex systems involve interactions between more than two agents. Hypergraphs capture these higher-order interactions through hyperedges that may link more than two nodes. We consider the problem of embedding a hypergraph into low-dimensional Euclidean space so that most interactions are short-range. This embedding is relevant to many follow-on tasks, such as node reordering, clustering, and visualization. We focus on two spectral embedding algorithms customized to hypergraphs which recover linear and periodic structures respectively. In the periodic case, nodes are positioned on the unit circle. We show that the two spectral hypergraph embedding algorithms are associated with a new class of generative hypergraph models. These models generate hyperedges according to node positions in the embedded space and encourage short-range connections. They allow us to quantify the relative presence of periodic and linear structures in the data through maximum likelihood. They also improve the interpretability of node embedding and provide a metric for hyperedge prediction. We demonstrate the hypergraph embedding and follow-on tasks-including quantifying relative strength of structures, clustering and hyperedge prediction-on synthetic and real-world hypergraphs. We find that the hypergraph approach can outperform clustering algorithms that use only dyadic edges. We also compare several triadic edge prediction methods on high school and primary school contact hypergraphs where our algorithm improves upon benchmark methods when the amount of training data is limited.

Order and disorder in the evolution of online knowledge community: an investigation of the chaotic behavior in social tagging systems with evidence of stack overflow

Purpose

Online question-and-answer (Q&A) communities serve as important channels for knowledge diffusion. The purpose of this study is to investigate the dynamic development process of online knowledge systems and explore the final or progressive state of system development. By measuring the nonlinear characteristics of knowledge systems from the perspective of complexity science, the authors aim to enrich the perspective and method of the research on the dynamics of knowledge systems, and to deeply understand the behavior rules of knowledge systems.

Design/methodology/approach

The authors collected data from the programming-related Q&A site Stack Overflow for a ten-year period (2008–2017) and included 48,373 tags in the analyses. The number of tags is taken as the time series, the correlation dimension and the maximum Lyapunov index are used to examine the chaos of the system and the Volterra series multistep forecast method is used to predict the system state.

Findings

There are strange attractors in the system, the whole system is complex but bounded and its evolution is bound to approach a relatively stable range. Empirical analyses indicate that chaos exists in the process of knowledge sharing in this social labeling system, and the period of change over time is about one week.

Originality/value

This study contributes to revealing the evolutionary cycle of knowledge stock in online knowledge systems and further indicates how this dynamic evolution can help in the setting of platform mechanics and resource inputs.

All Models Are Wrong, and Some Are Religious: Supernatural Explanations as Abstract and Useful Falsehoods about Complex Realities

Many cognitive and evolutionary theories of religion argue that supernatural explanations are byproducts of our cognitive adaptations. An influential argument states that our supernatural explanations result from a tendency to generate anthropomorphic explanations, and that this tendency is a byproduct of an error management strategy because agents tend to be associated with especially high fitness costs. We propose instead that anthropomorphic and other supernatural explanations result as features of a broader toolkit of well-designed cognitive adaptations, which are designed for explaining the abstract and causal structure of complex, unobservable, and uncertain phenomena that have substantial impacts on fitness. Specifically, we argue that (1) mental representations about the abstract vs. the supernatural are largely overlapping, if not identical, and (2) when the data-generating processes for scarce and ambiguous observations are complex and opaque, a naive observer can improve a bias-variance trade-off by starting with a simple, underspecified explanation that Western observers readily interpret as "supernatural." We then argue that (3) in many cases, knowledge specialists across cultures offer pragmatic services that involve apparently supernatural explanations, and their clients are frequently willing to pay them in a market for useful and effective services. We propose that at least some ethnographic descriptions of religion might actually reflect ordinary and adaptive responses to novel problems such as illnesses and natural disasters, where knowledge specialists possess and apply the best available explanations about phenomena that would otherwise be completely mysterious and unpredictable.

Inference of hyperedges and overlapping communities in hypergraphs

Hypergraphs, encoding structured interactions among any number of system units, have recently proven a successful tool to describe many real-world biological and social networks. Here we propose a framework based on statistical inference to characterize the structural organization of hypergraphs. The method allows to infer missing hyperedges of any size in a principled way, and to jointly detect overlapping communities in presence of higher-order interactions. Furthermore, our model has an efficient numerical implementation, and it runs faster than dyadic algorithms on pairwise records projected from higher-order data. We apply our method to a variety of real-world systems, showing strong performance in hyperedge prediction tasks, detecting communities well aligned with the information carried by interactions, and robustness against addition of noisy hyperedges. Our approach illustrates the fundamental advantages of a hypergraph probabilistic model when modeling relational systems with higher-order interactions.

Coexistence in diverse communities with higher-order interactions

A central assumption in most ecological models is that the interactions in a community operate only between pairs of species. However, two species may interactively affect the growth of a focal species. Although interactions among three or more species, called higher-order interactions, have the potential to modify our theoretical understanding of coexistence, ecologists lack clear expectations for how these interactions shape community structure. Here we analytically predict and numerically confirm how the variability and strength of higher-order interactions affect species coexistence. We found that as higher-order interaction strengths became more variable across species, fewer species could coexist, echoing the behavior of pairwise models. If interspecific higher-order interactions became too harmful relative to self-regulation, coexistence in diverse communities was destabilized, but coexistence was also lost when these interactions were too weak and mutualistic higher-order effects became prevalent. This behavior depended on the functional form of the interactions as the destabilizing effects of the mutualistic higher-order interactions were ameliorated when their strength saturated with species' densities. Last, we showed that more species-rich communities structured by higher-order interactions lose species more readily than their species-poor counterparts, generalizing classic results for community stability. Our work provides needed theoretical expectations for how higher-order interactions impact species coexistence in diverse communities.

Understanding Hazardous Materials Transportation Accidents Based on Higher-Order Network Theory

In hazardous materials transportation systems, accident causation analysis is important to transportation safety. Complex network theory can be effectively used to understand the causal factors of and their relationships within accidents. In this paper, a higher-order network method is proposed to establish a hazardous materials transportation accident causation network (HMTACN), which considers the sequences and dependences of causal factors. The HMTACN is composed of 125 first- and 118 higher-order nodes that represent causes, and 545 directed edges that denote complex relationships among causes. By analyzing topological properties, the results show that the HMTACN has the characteristics of small-world networks and displays the properties of scale-free networks. Additionally, critical causal factors and key relationships of the HMTACN are discovered. Moreover, unsafe tank or valve states are important causal factors; and leakage, roll-over, collision, and fire are most likely to trigger chain reactions. Important higher-order nodes are discovered, which can represent key relationships in the HMTACN. For example, unsafe distance and improper operation usually lead to collision and roll-over. These results of higher-order nodes cannot be found by the traditional Markov network model. This study provides a practical way to extract and construct an accident causation network from numerous accident investigation reports. It also provides insights into safety management of hazardous materials transportation.

Controlling species densities in structurally perturbed intransitive cycles with higher-order interactions

The persistence of biodiversity of species is a challenging proposition in ecological communities in the face of Darwinian selection. The present article investigates beyond the pairwise competitive interactions and provides a novel perspective for understanding the influence of higher-order interactions on the evolution of social phenotypes. Our simple model yields a prosperous outlook to demonstrate the impact of perturbations on intransitive competitive higher-order interactions. Using a mathematical technique, we show how alone the perturbed interaction network can quickly determine the coexistence equilibrium of competing species instead of solving a large system of ordinary differential equations. It is possible to split the system into multiple feasible cluster states depending on the number of perturbations. Our analysis also reveals that the ratio between the unperturbed and perturbed species is inversely proportional to the amount of employed perturbation. Our results suggest that nonlinear dynamical systems and interaction topologies can be interplayed to comprehend species' coexistence under adverse conditions. Particularly, our findings signify that less competition between two species increases their abundance and outperforms others.

Impact of basic network motifs on the collective response to perturbations

Many collective phenomena such as epidemic spreading and cascading failures in socioeconomic systems on networks are caused by perturbations of the dynamics. How perturbations propagate through networks, impact and disrupt their functions may depend on the network, the type and location of the perturbation as well as the spreading dynamics. Previous work has analyzed the retardation effects of the nodes along the propagation paths, suggesting a few transient propagation "scaling” regimes as a function of the nodes’ degree, but regardless of motifs such as triangles. Yet, empirical networks consist of motifs enabling the proper functioning of the system. Here, we show that basic motifs along the propagation path jointly determine the previously proposed scaling regimes of distance-limited propagation and degree-limited propagation, or even cease their existence. Our results suggest a radical departure from these scaling regimes and provide a deeper understanding of the interplay of self-dynamics, interaction dynamics, and topological properties.

Spreading processes and cascading failures on complex networks are often triggered by external perturbations. The authors uncover the impact of network motifs on the processes of perturbations propagation through networks, and networks’ response dynamics.

Dynamical systems defined on simplicial complexes: Symmetries, conjugacies, and invariant subspaces

We consider the general model for dynamical systems defined on a simplicial complex. We describe the conjugacy classes of these systems and show how symmetries in a given simplicial complex manifest in the dynamics defined thereon, especially with regard to invariant subspaces in the dynamics.

MSDA-NMF: A Multilayer Complex System Model Integrating Deep Autoencoder and NMF

In essence, the network is a way of encoding the information of the underlying social management system. Ubiquitous social management systems rarely exist alone and have dynamic complexity. For complex social management systems, it is difficult to extract and represent multi-angle features of data only by using non-negative matrix factorization. Existing deep NMF models integrating multi-layer information struggle to explain the results obtained after mid-layer NMF. In this paper, NMF is introduced into the multi-layer NMF structure, and the feature representation of the input data is realized by using the complex hierarchical structure. By adding regularization constraints for each layer, the essential features of the data are obtained by characterizing the feature transformation layer-by-layer. Furthermore, the deep autoencoder and NMF are fused to construct the multi-layer NMF model MSDA-NMF that integrates the deep autoencoder. Through multiple data sets such as HEP-TH, OAG and HEP-TH, Pol blog, Orkut and Livejournal, compared with 8 popular NMF models, the Micro index of the better model increased by 1.83, NMI value increased by 12%, and link prediction performance improved by 13%. Furthermore, the robustness of the proposed model is verified.

What Models and Tools can Contribute to a Better Understanding of Brain Activity?

Despite impressive scientific advances in understanding the structure and function of the human brain, big challenges remain. A deep understanding of healthy and aberrant brain activity at a wide range of temporal and spatial scales is needed. Here we discuss, from an interdisciplinary network perspective, the advancements in physical and mathematical modeling as well as in data analysis techniques that, in our opinion, have potential to further advance our understanding of brain structure and function.

Spatial evolution of cooperation with variable payoffs

In the evolution of cooperation, the individuals' payoffs are commonly random in real situations, e.g., the social networks and the economic regions, leading to unpredictable factors. Therefore, there are chances for each individual to obtain the exceeding payoff and risks to get the low payoff. In this paper, we consider that each individual's payoff follows a specific probability distribution with a fixed expectation, where the normal distribution and the exponential distribution are employed in our model. In the simulations, we perform the models on the weak prisoner's dilemmas (WPDs) and the snowdrift games (SDGs), and four types of networks, including the hexagon lattice, the square lattice, the small-world network, and the triangular lattice are considered. For the individuals' normally distributed payoff, we find that the higher standard deviation usually inhibits the cooperation for the WPDs but promotes the cooperation for the SDGs. Besides, with a higher standard deviation, the cooperation clusters are usually split for the WPDs but constructed for the SDGs. For the individuals' exponentially distributed payoff, we find that the small-world network provides the best condition for the emergence of cooperators in WPDs and SDGs. However, when playing SDGs, the small-world network allows the smallest space for the pure cooperative state while the hexagon lattice allows the largest.