Inference of hyperedges and overlapping communities in hypergraphs

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.

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.

Stochastic block hypergraph model

The stochastic block model is widely used to generate graphs with a community structure, but no simple alternative currently exists for hypergraphs, in which more than two nodes can be connected together through a hyperedge. We discuss here such a hypergraph generalization, based on the clustering connection probability P_{ij} between nodes of communities i and j, and that uses an explicit and modulable hyperedge formation process. We focus on the standard case where P_{ij}=pδ_{ij}+q(1-δ_{ij}) when 0≤q≤p (δ_{ij} is the Kronecker symbol). We propose a simple model that satisfies three criteria: it should be as simple as possible, when p=q the model should be equivalent to the standard hypergraph random model, and it should use an explicit and modulable hyperedge formation process so that the model is intuitive and can easily express different real-world formation processes. We first show that for such a model the degree distribution and hyperedge size distribution can be approximated by binomial distributions with effective parameters that depend on the number of communities and q/p. Also, the composition of hyperedges goes for q=0 from 'pure' hyperedges (comprising nodes belonging to the same community) to 'mixed' hyperedges that comprise nodes from different communities for q=p. We test various formation processes and our results suggest that when they depend on the composition of the hyperedge, they tend to favor the dominant community and lead to hyperedges with a smaller diversity. In contrast, for formation processes that are independent from the hyperedge structure, we obtain hyperedges comprising a larger diversity of communities. The advantages of the model proposed here are its simplicity and flexibility that make it a good candidate for testing community-related problems, such as their detection, impact on various dynamics, and visualization.

Structure and inference in hypergraphs with node attributes

Many networked datasets with units interacting in groups of two or more, encoded with hypergraphs, are accompanied by extra information about nodes, such as the role of an individual in a workplace. Here we show how these node attributes can be used to improve our understanding of the structure resulting from higher-order interactions. We consider the problem of community detection in hypergraphs and develop a principled model that combines higher-order interactions and node attributes to better represent the observed interactions and to detect communities more accurately than using either of these types of information alone. The method learns automatically from the input data the extent to which structure and attributes contribute to explain the data, down weighing or discarding attributes if not informative. Our algorithmic implementation is efficient and scales to large hypergraphs and interactions of large numbers of units. We apply our method to a variety of systems, showing strong performance in hyperedge prediction tasks and in selecting community divisions that correlate with attributes when these are informative, but discarding them otherwise. Our approach illustrates the advantage of using informative node attributes when available with higher-order data.

The Ihara zeta function as a partition function for network structure characterisation

Statistical characterizations of complex network structures can be obtained from both the Ihara Zeta function (in terms of prime cycle frequencies) and the partition function from statistical mechanics. However, these two representations are usually regarded as separate tools for network analysis, without exploiting the potential synergies between them. In this paper, we establish a link between the Ihara Zeta function from algebraic graph theory and the partition function from statistical mechanics, and exploit this relationship to obtain a deeper structural characterisation of network structure. Specifically, the relationship allows us to explore the connection between the microscopic structure and the macroscopic characterisation of a network. We derive thermodynamic quantities describing the network, such as entropy, and show how these are related to the frequencies of prime cycles of various lengths. In particular, the n-th order partial derivative of the Ihara Zeta function can be used to compute the number of prime cycles in a network, which in turn is related to the partition function of Bose-Einstein statistics. The corresponding derived entropy allows us to explore a phase transition in the network structure with critical points at high and low-temperature limits. Numerical experiments and empirical data are presented to evaluate the qualitative and quantitative performance of the resulting structural network characterisations.

Higher-order correlations reveal complex memory in temporal hypergraphs

Many real-world complex systems are characterized by interactions in groups that change in time. Current temporal network approaches, however, are unable to describe group dynamics, as they are based on pairwise interactions only. Here, we use time-varying hypergraphs to describe such systems, and we introduce a framework based on higher-order correlations to characterize their temporal organization. The analysis of human interaction data reveals the existence of coherent and interdependent mesoscopic structures, thus capturing aggregation, fragmentation and nucleation processes in social systems. We introduce a model of temporal hypergraphs with non-Markovian group interactions, which reveals complex memory as a fundamental mechanism underlying the emerging pattern in the data.

Advancing urban traffic accident forecasting through sparse spatio-temporal dynamic learning

Traffic accidents have emerged as one of the most public health safety matters, raising concerns from both the public and urban administrators. The ability to accurately predict traffic accident not only supports the governmental decision-making in advance but also enhances public confidence in safety measures. However, the efficacy of traditional spatio-temporal prediction models are compromised by the skewed distributions and sparse labeling of accident data. To this end, we propose a Sparse Spatio-Temporal Dynamic Hypergraph Learning (SST-DHL) framework that captures higher-order dependencies in sparse traffic accidents by combining hypergraph learning and self-supervised learning. The SST-DHL model incorporates a multi-view spatiotemporal convolution block to capture local correlations and semantics of traffic accidents, a cross-regional dynamic hypergraph learning model to identify global spatiotemporal dependencies, and a two-supervised self-learning paradigm to capture both local and global spatiotemporal patterns. Through experimentation on New York City and London accident datasets, we demonstrate that our proposed SST-DHL exhibits significant improvements compared to optimal baseline models at different sparsity levels. Additionally, it offers enhanced interpretability of results by elucidating complex spatio-temporal dependencies among various traffic accident instances. Our study demonstrates the effectiveness of the SST-DHL framework in accurately predicting traffic accidents, thereby enhancing public safety and trust.

A blueprint for a new commercial driving epidemiology: An emerging paradigm grounded in integrative exposome and network epistemologies

Excess health and safety risks of commercial drivers are largely determined by, embedded in, or operate as complex, dynamic, and randomly determined systems with interacting parts. Yet, prevailing epidemiology is entrenched in narrow, deterministic, and static exposure-response frameworks along with ensuing inadequate data and limiting methods, thereby perpetuating an incomplete understanding of commercial drivers' health and safety risks. This paper is grounded in our ongoing research that conceptualizes health and safety challenges of working people as multilayered "wholes" of interacting work and nonwork factors, exemplified by complex-systems epistemologies. Building upon and expanding these assumptions, herein we: (a) discuss how insights from integrative exposome and network-science-based frameworks can enhance our understanding of commercial drivers' chronic disease and injury burden; (b) introduce the "working life exposome of commercial driving" (WLE-CD)-an array of multifactorial and interdependent work and nonwork exposures and associated biological responses that concurrently or sequentially impact commercial drivers' health and safety during and beyond their work tenure; (c) conceptualize commercial drivers' health and safety risks as multilayered networks centered on the WLE-CD and network relational patterns and topological properties-that is, arrangement, connections, and relationships among network components-that largely govern risk dynamics; and (d) elucidate how integrative exposome and network-science-based innovations can contribute to a more comprehensive understanding of commercial drivers' chronic disease and injury risk dynamics. Development, validation, and proliferation of this emerging discourse can move commercial driving epidemiology to the frontier of science with implications for policy, action, other working populations, and population health at large.

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'.

Patterns in Temporal Networks with Higher-Order Egocentric Structures

The analysis of complex and time-evolving interactions, such as those within social dynamics, represents a current challenge in the science of complex systems. Temporal networks stand as a suitable tool for schematizing such systems, encoding all the interactions appearing between pairs of individuals in discrete time. Over the years, network science has developed many measures to analyze and compare temporal networks. Some of them imply a decomposition of the network into small pieces of interactions; i.e., only involving a few nodes for a short time range. Along this line, a possible way to decompose a network is to assume an egocentric perspective; i.e., to consider for each node the time evolution of its neighborhood. This was proposed by Longa et al. by defining the "egocentric temporal neighborhood", which has proven to be a useful tool for characterizing temporal networks relative to social interactions. However, this definition neglects group interactions (quite common in social domains), as they are always decomposed into pairwise connections. A more general framework that also allows considering larger interactions is represented by higher-order networks. Here, we generalize the description of social interactions to hypergraphs. Consequently, we generalize their decomposition into "hyper egocentric temporal neighborhoods". This enables the analysis of social interactions, facilitating comparisons between different datasets or nodes within a dataset, while considering the intrinsic complexity presented by higher-order interactions. Even if we limit the order of interactions to the second order (triplets of nodes), our results reveal the importance of a higher-order representation.In fact, our analyses show that second-order structures are responsible for the majority of the variability at all scales: between datasets, amongst nodes, and over time.

Framework to generate hypergraphs with community structure

In recent years hypergraphs have emerged as a powerful tool to study systems with multibody interactions which cannot be trivially reduced to pairs. While highly structured methods to generate synthetic data have proved fundamental for the standardized evaluation of algorithms and the statistical study of real-world networked data, these are scarcely available in the context of hypergraphs. Here we propose a flexible and efficient framework for the generation of hypergraphs with many nodes and large hyperedges, which allows specifying general community structures and tune different local statistics. We illustrate how to use our model to sample synthetic data with desired features (assortative or disassortative communities, mixed or hard community assignments, etc.), analyze community detection algorithms, and generate hypergraphs structurally similar to real-world data. Overcoming previous limitations on the generation of synthetic hypergraphs, our work constitutes a substantial advancement in the statistical modeling of higher-order systems.

The time-evolving epileptic brain network: concepts, definitions, accomplishments, perspectives

Epilepsy is now considered a network disease that affects the brain across multiple levels of spatial and temporal scales. The paradigm shift from an epileptic focus-a discrete cortical area from which seizures originate-to a widespread epileptic network-spanning lobes and hemispheres-considerably advanced our understanding of epilepsy and continues to influence both research and clinical treatment of this multi-faceted high-impact neurological disorder. The epileptic network, however, is not static but evolves in time which requires novel approaches for an in-depth characterization. In this review, we discuss conceptual basics of network theory and critically examine state-of-the-art recording techniques and analysis tools used to assess and characterize a time-evolving human epileptic brain network. We give an account on current shortcomings and highlight potential developments towards an improved clinical management of epilepsy.

Hyperedge prediction and the statistical mechanisms of higher-order and lower-order interactions in complex networks

Complex networked systems often exhibit higher-order interactions, beyond dyadic interactions, which can dramatically alter their observed behavior. Consequently, understanding hypergraphs from a structural perspective has become increasingly important. Statistical, group-based inference approaches are well suited for unveiling the underlying community structure and predicting unobserved interactions. However, these approaches often rely on two key assumptions: that the same groups can explain hyperedges of any order and that interactions are assortative, meaning that edges are formed by nodes with the same group memberships. To test these assumptions, we propose a group-based generative model for hypergraphs that does not impose an assortative mechanism to explain observed higher-order interactions, unlike current approaches. Our model allows us to explore the validity of the assumptions. Our results indicate that the first assumption appears to hold true for real networks. However, the second assumption is not necessarily accurate; we find that a combination of general statistical mechanisms can explain observed hyperedges. Finally, with our approach, we are also able to determine the importance of lower and high-order interactions for predicting unobserved interactions. Our research challenges the conventional assumptions of group-based inference methodologies and broadens our understanding of the underlying structure of hypergraphs.

Partial entropy decomposition reveals higher-order information structures in human brain activity

The standard approach to modeling the human brain as a complex system is with a network, where the basic unit of interaction is a pairwise link between two brain regions. While powerful, this approach is limited by the inability to assess higher-order interactions involving three or more elements directly. In this work, we explore a method for capturing higher-order dependencies in multivariate data: the partial entropy decomposition (PED). Our approach decomposes the joint entropy of the whole system into a set of nonnegative atoms that describe the redundant, unique, and synergistic interactions that compose the system's structure. PED gives insight into the mathematics of functional connectivity and its limitation. When applied to resting-state fMRI data, we find robust evidence of higher-order synergies that are largely invisible to standard functional connectivity analyses. Our approach can also be localized in time, allowing a frame-by-frame analysis of how the distributions of redundancies and synergies change over the course of a recording. We find that different ensembles of regions can transiently change from being redundancy-dominated to synergy-dominated and that the temporal pattern is structured in time. These results provide strong evidence that there exists a large space of unexplored structures in human brain data that have been largely missed by a focus on bivariate network connectivity models. This synergistic structure is dynamic in time and likely will illuminate interesting links between brain and behavior. Beyond brain-specific application, the PED provides a very general approach for understanding higher-order structures in a variety of complex systems.

Community detection in large hypergraphs

Hypergraphs, describing networks where interactions take place among any number of units, are a natural tool to model many real-world social and biological systems. Here, we propose a principled framework to model the organization of higher-order data. Our approach recovers community structure with accuracy exceeding that of currently available state-of-the-art algorithms, as tested in synthetic benchmarks with both hard and overlapping ground-truth partitions. Our model is flexible and allows capturing both assortative and disassortative community structures. Moreover, our method scales orders of magnitude faster than competing algorithms, making it suitable for the analysis of very large hypergraphs, containing millions of nodes and interactions among thousands of nodes. Our work constitutes a practical and general tool for hypergraph analysis, broadening our understanding of the organization of real-world higher-order systems.

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.