Social contagion models on hypergraphs

Dynamics of simplicial SEIRS epidemic model: global asymptotic stability and neural Lyapunov functions

The transmission of infectious diseases on a particular network is ubiquitous in the physical world. Here, we investigate the transmission mechanism of infectious diseases with an incubation period using a networked compartment model that contains simplicial interactions, a typical high-order structure. We establish a simplicial SEIRS model and find that the proportion of infected individuals in equilibrium increases due to the many-body connections, regardless of the type of connections used. We analyze the dynamics of the established model, including existence and local asymptotic stability, and highlight differences from existing models. Significantly, we demonstrate global asymptotic stability using the neural Lyapunov function, a machine learning technique, with both numerical simulations and rigorous analytical arguments. We believe that our model owns the potential to provide valuable insights into transmission mechanisms of infectious diseases on high-order network structures, and that our approach and theory of using neural Lyapunov functions to validate model asymptotic stability can significantly advance investigations on complex dynamics of infectious disease.

Learning the effective order of a hypergraph dynamical system

Dynamical systems on hypergraphs can display a rich set of behaviors not observable for systems with pairwise interactions. Given a distributed dynamical system with a putative hypergraph structure, an interesting question is thus how much of this hypergraph structure is actually necessary to faithfully replicate the observed dynamical behavior. To answer this question, we propose a method to determine the minimum order of a hypergraph necessary to approximate the corresponding dynamics accurately. Specifically, we develop a mathematical framework that allows us to determine this order when the type of dynamics is known. We use these ideas in conjunction with a hypergraph neural network to directly learn the dynamics itself and the resulting order of the hypergraph from both synthetic and real datasets consisting of observed system trajectories.

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.

Stochastic modeling of cascade dynamics: A unified approach for simple and complex contagions across homogeneous and heterogeneous threshold distributions on networks

The COVID-19 pandemic has underscored the importance of understanding, forecasting, and avoiding infectious processes, as well as the necessity for understanding the diffusion and acceptance of preventative measures. Simple contagions, like virus transmission, can spread with a single encounter, while complex contagions, such as preventive social measures (e.g., wearing masks, social distancing), may require multiple interactions to propagate. This disparity in transmission mechanisms results in differing contagion rates and contagion patterns between viruses and preventive measures. Furthermore, the dynamics of complex contagions are significantly less understood than those of simple contagions. Stochastic models, integrating inherent variability and randomness, offer a way to elucidate complex contagion dynamics. This paper introduces a stochastic model for both simple and complex contagions and assesses its efficacy against ensemble simulations for homogeneous and heterogeneous threshold configurations. The model provides a unified framework for analyzing both types of contagions, demonstrating promising outcomes across various threshold setups on Erds-Rényi graphs.

Higher-Order Components Dictate Higher-Order Contagion Dynamics in Hypergraphs

The presence of the giant component is a necessary condition for the emergence of collective behavior in complex networked systems. Unlike networks, hypergraphs have an important native feature that components of hypergraphs might be of higher order, which could be defined in terms of the number of common nodes shared between hyperedges. Although the extensive higher-order component (HOC) could be witnessed ubiquitously in real-world hypergraphs, the role of the giant HOC in collective behavior on hypergraphs has yet to be elucidated. In this Letter, we demonstrate that the presence of the giant HOC fundamentally alters the outbreak patterns of higher-order contagion dynamics on real-world hypergraphs. Most crucially, the giant HOC is required for the higher-order contagion to invade globally from a single seed. We confirm it by using synthetic random hypergraphs containing adjustable and analytically calculable giant HOC.

Phase chimera states on nonlocal hyperrings

Chimera states are dynamical states where regions of synchronous trajectories coexist with incoherent ones. A significant amount of research has been devoted to studying chimera states in systems of identical oscillators, nonlocally coupled through pairwise interactions. Nevertheless, there is increasing evidence, also supported by available data, that complex systems are composed of multiple units experiencing many-body interactions that can be modeled by using higher-order structures beyond the paradigm of classic pairwise networks. In this work we investigate whether phase chimera states appear in this framework, by focusing on a topology solely involving many-body, nonlocal, and nonregular interactions, hereby named nonlocal d-hyperring, (d+1) being the order of the interactions. We present the theory by using the paradigmatic Stuart-Landau oscillators as node dynamics, and we show that phase chimera states emerge in a variety of structures and with different coupling functions. For comparison, we show that, when higher-order interactions are "flattened" to pairwise ones, the chimera behavior is weaker and more elusive.

Probabilistic activity driven model of temporal simplicial networks and its application on higher-order dynamics

Network modeling characterizes the underlying principles of structural properties and is of vital significance for simulating dynamical processes in real world. However, bridging structure and dynamics is always challenging due to the multiple complexities in real systems. Here, through introducing the individual's activity rate and the possibility of group interaction, we propose a probabilistic activity-driven (PAD) model that could generate temporal higher-order networks with both power-law and high-clustering characteristics, which successfully links the two most critical structural features and a basic dynamical pattern in extensive complex systems. Surprisingly, the power-law exponents and the clustering coefficients of the aggregated PAD network could be tuned in a wide range by altering a set of model parameters. We further provide an approximation algorithm to select the proper parameters that can generate networks with given structural properties, the effectiveness of which is verified by fitting various real-world networks. Finally, we construct the co-evolution framework of the PAD model and higher-order contagion dynamics and derive the critical conditions for phase transition and bistable phenomenon using theoretical and numerical methods. Results show that tendency of participating in higher-order interactions can promote the emergence of bistability but delay the outbreak under heterogeneous activity rates. Our model provides a basic tool to reproduce complex structural properties and to study the widespread higher-order dynamics, which has great potential for applications across fields.

Nonlinear bias toward complex contagion in uncertain transmission settings

Current epidemics in the biological and social domains are challenging the standard assumptions of mathematical contagion models. Chief among them are the complex patterns of transmission caused by heterogeneous group sizes and infection risk varying by orders of magnitude in different settings, like indoor versus outdoor gatherings in the COVID-19 pandemic or different moderation practices in social media communities. However, quantifying these heterogeneous levels of risk is difficult, and most models typically ignore them. Here, we include these features in an epidemic model on weighted hypergraphs to capture group-specific transmission rates. We study analytically the consequences of ignoring the heterogeneous transmissibility and find an induced superlinear infection rate during the emergence of a new outbreak, even though the underlying mechanism is a simple, linear contagion. The dynamics produced at the individual and group levels are therefore more similar to complex, nonlinear contagions, thus blurring the line between simple and complex contagions in realistic settings. We support this claim by introducing a Bayesian inference framework to quantify the nonlinearity of contagion processes. We show that simple contagions on real weighted hypergraphs are systematically biased toward the superlinear regime if the heterogeneity of the weights is ignored, greatly increasing the risk of erroneous classification as complex contagions. Our results provide an important cautionary tale for the challenging task of inferring transmission mechanisms from incidence data. Yet, it also paves the way for effective models that capture complex features of epidemics through nonlinear infection rates.

Nature of hypergraph k-core percolation problems

Hypergraphs are higher-order networks that capture the interactions between two or more nodes. Hypergraphs can always be represented by factor graphs, i.e., bipartite networks between nodes and factor nodes (representing groups of nodes). Despite this universal representation, here we reveal that k-core percolation on hypergraphs can be significantly distinct from k-core percolation on factor graphs. We formulate the theory of hypergraph k-core percolation based on the assumption that a hyperedge can be intact only if all its nodes are intact. This scenario applies, for instance, to supply chains where the production of a product requires all raw materials and all processing steps; in biology it applies to protein-interaction networks where protein complexes can function only if all the proteins are present; and it applies as well to chemical reaction networks where a chemical reaction can take place only when all the reactants are present. Formulating a message-passing theory for hypergraph k-core percolation, and combining it with the theory of critical phenomena on networks, we demonstrate sharp differences with previously studied factor graph k-core percolation processes where it is allowed for hyperedges to have one or more damaged nodes and still be intact. To solve the dichotomy between k-core percolation on hypegraphs and on factor graphs, we define a set of pruning processes that act either exclusively on nodes or exclusively on hyperedges and depend on their second-neighborhood connectivity. We show that the resulting second-neighbor k-core percolation problems are significantly distinct from each other. Moreover we reveal that although these processes remain distinct from factor graphs k-core processes, when the pruning process acts exclusively on hyperedges the phase diagram is reduced to the one of factor graph k-cores.

Robustness and Complexity of Directed and Weighted Metabolic Hypergraphs

Metabolic networks are probably among the most challenging and important biological networks. Their study provides insight into how biological pathways work and how robust a specific organism is against an environment or therapy. Here, we propose a directed hypergraph with edge-dependent vertex weight as a novel framework to represent metabolic networks. This hypergraph-based representation captures higher-order interactions among metabolites and reactions, as well as the directionalities of reactions and stoichiometric weights, preserving all essential information. Within this framework, we propose the communicability and the search information as metrics to quantify the robustness and complexity of directed hypergraphs. We explore the implications of network directionality on these measures and illustrate a practical example by applying them to a small-scale E. coli core model. Additionally, we compare the robustness and the complexity of 30 different models of metabolism, connecting structural and biological properties. Our findings show that antibiotic resistance is associated with high structural robustness, while the complexity can distinguish between eukaryotic and prokaryotic organisms.

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.

Contagion dynamics on hypergraphs with nested hyperedges

In complex social systems encoded as hypergraphs, higher-order (i.e., group) interactions taking place among more than two individuals are represented by hyperedges. One of the higher-order correlation structures native to hypergraphs is the nestedness: Some hyperedges can be entirely contained (that is, nested) within another larger hyperedge, which itself can also be nested further in a hierarchical manner. Yet the effect of such hierarchical structure of hyperedges on the dynamics has remained unexplored. In this context, here we propose a random nested-hypergraph model with a tunable level of nestedness and investigate the effects of nestedness on a higher-order susceptible-infected-susceptible process. By developing an analytic framework called the facet approximation, we obtain the steady-state fraction of infected nodes on the random nested-hypergraph model more accurately than existing methods. Our results show that the hyperedge-nestedness affects the phase diagram significantly. Monte Carlo simulations support the analytical results.

Identifying Vital Nodes in Hypergraphs Based on Von Neumann Entropy

Hypergraphs have become an accurate and natural expression of high-order coupling relationships in complex systems. However, applying high-order information from networks to vital node identification tasks still poses significant challenges. This paper proposes a von Neumann entropy-based hypergraph vital node identification method (HVC) that integrates high-order information as well as its optimized version (semi-SAVC). HVC is based on the high-order line graph structure of hypergraphs and measures changes in network complexity using von Neumann entropy. It integrates s-line graph information to quantify node importance in the hypergraph by mapping hyperedges to nodes. In contrast, semi-SAVC uses a quadratic approximation of von Neumann entropy to measure network complexity and considers only half of the maximum order of the hypergraph's s-line graph to balance accuracy and efficiency. Compared to the baseline methods of hyperdegree centrality, closeness centrality, vector centrality, and sub-hypergraph centrality, the new methods demonstrated superior identification of vital nodes that promote the maximum influence and maintain network connectivity in empirical hypergraph data, considering the influence and robustness factors. The correlation and monotonicity of the identification results were quantitatively analyzed and comprehensive experimental results demonstrate the superiority of the new methods. At the same time, a key non-trivial phenomenon was discovered: influence does not increase linearly as the s-line graph orders increase. We call this the saturation effect of high-order line graph information in hypergraph node identification. When the order reaches its saturation value, the addition of high-order information often acts as noise and affects propagation.

Higher-order interactions in Kuramoto oscillators with inertia

How do higher-order interactions influence the dynamical landscape of a network of the second-order phase oscillators? We address this question using three coupled Kuramoto phase oscillators with inertia under pairwise and higher-order interactions, in search of various collective states, and new states, if any, that show marginal presence or absence under pairwise interactions. We explore this small network for varying phase lag in the coupling and over a range of negative to positive coupling strength of pairwise as well as higher-order or group interactions. In the extended coupling parameter plane of the network we record several well-known states such as synchronization, frequency chimera states, and rotating waves that appear with distinct boundaries. In the parameter space, we also find states generated by the influence of higher-order interactions: The 2+1 antipodal point and the 2+1 phase-locked states. Our results demonstrate the importantance of the choices of the phase lag and the sign of the higher-order coupling strength for the emergent dynamics of the network. We provide analytical support to our numerical results.

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.

Dynamical Analysis of Hyper-ILSR Rumor Propagation Model with Saturation Incidence Rate

With the development of the Internet, it is more convenient for people to obtain information, which also facilitates the spread of rumors. It is imperative to study the mechanisms of rumor transmission to control the spread of rumors. The process of rumor propagation is often affected by the interaction of multiple nodes. To reflect higher-order interactions in rumor-spreading, hypergraph theories are introduced in a Hyper-ILSR (Hyper-Ignorant-Lurker-Spreader-Recover) rumor-spreading model with saturation incidence rate in this study. Firstly, the definition of hypergraph and hyperdegree is introduced to explain the construction of the model. Secondly, the existence of the threshold and equilibrium of the Hyper-ILSR model is revealed by discussing the model, which is used to judge the final state of rumor propagation. Next, the stability of equilibrium is studied by Lyapunov functions. Moreover, optimal control is put forward to suppress rumor propagation. Finally, the differences between the Hyper-ILSR model and the general ILSR model are shown in numerical simulations.

Dimension reduction in higher-order contagious phenomena

We investigate epidemic spreading in a deterministic susceptible-infected-susceptible model on uncorrelated heterogeneous networks with higher-order interactions. We provide a recipe for the construction of one-dimensional reduced model (resilience function) of the N-dimensional susceptible-infected-susceptible dynamics in the presence of higher-order interactions. Utilizing this reduction process, we are able to capture the microscopic and macroscopic behavior of infectious networks. We find that the microscopic state of nodes (fraction of stable healthy individual of each node) inversely scales with their degree, and it becomes diminished due to the presence of higher-order interactions. In this case, we analytically obtain that the macroscopic state of the system (fraction of infectious or healthy population) undergoes abrupt transition. Additionally, we quantify the network's resilience, i.e., how the topological changes affect the stable infected population. Finally, we provide an alternative framework of dimension reduction based on the spectral analysis of the network, which can identify the critical onset of the disease in the presence or absence of higher-order interactions. Both reduction methods can be extended for a large class of dynamical models.

Multistability, intermittency, and hybrid transitions in social contagion models on hypergraphs

Although ubiquitous, interactions in groups of individuals are not yet thoroughly studied. Frequently, single groups are modeled as critical-mass dynamics, which is a widespread concept used not only by academics but also by politicians and the media. However, less explored questions are how a collection of groups will behave and how their intersection might change the dynamics. Here, we formulate this process as binary-state dynamics on hypergraphs. We showed that our model has a rich behavior beyond discontinuous transitions. Notably, we have multistability and intermittency. We demonstrated that this phenomenology could be associated with community structures, where we might have multistability or intermittency by controlling the number or size of bridges between communities. Furthermore, we provided evidence that the observed transitions are hybrid. Our findings open new paths for research, ranging from physics, on the formal calculation of quantities of interest, to social sciences, where new experiments can be designed.

The dynamic nature of percolation on networks with triadic interactions

Percolation establishes the connectivity of complex networks and is one of the most fundamental critical phenomena for the study of complex systems. On simple networks, percolation displays a second-order phase transition; on multiplex networks, the percolation transition can become discontinuous. However, little is known about percolation in networks with higher-order interactions. Here, we show that percolation can be turned into a fully fledged dynamical process when higher-order interactions are taken into account. By introducing signed triadic interactions, in which a node can regulate the interactions between two other nodes, we define triadic percolation. We uncover that in this paradigmatic model the connectivity of the network changes in time and that the order parameter undergoes a period doubling and a route to chaos. We provide a general theory for triadic percolation which accurately predicts the full phase diagram on random graphs as confirmed by extensive numerical simulations. We find that triadic percolation on real network topologies reveals a similar phenomenology. These results radically change our understanding of percolation and may be used to study complex systems in which the functional connectivity is changing in time dynamically and in a non-trivial way, such as in neural and climate networks.

Eigenvector localization in hypergraphs: Pairwise versus higher-order links

Localization behaviors of Laplacian eigenvectors of complex networks furnish an explanation to various dynamical phenomena of the corresponding complex systems. We numerically examine roles of higher-order and pairwise links in driving eigenvector localization of hypergraphs Laplacians. We find that pairwise interactions can engender localization of eigenvectors corresponding to small eigenvalues for some cases, whereas higher-order interactions, even being much much less than the pairwise links, keep steering localization of the eigenvectors corresponding to larger eigenvalues for all the cases considered here. These results will be advantageous to comprehend dynamical phenomena, such as diffusion, and random walks on a range of real-world complex systems having higher-order interactions in better manner.

Aging in binary-state models: The Threshold model for complex contagion

We study the non-Markovian effects associated with aging for binary-state dynamics in complex networks. Aging is considered as the property of the agents to be less prone to change their state the longer they have been in the current state, which gives rise to heterogeneous activity patterns. In particular, we analyze aging in the Threshold model, which has been proposed to explain the process of adoption of new technologies. Our analytical approximations give a good description of extensive Monte Carlo simulations in Erdős-Rényi, random-regular and Barabási-Albert networks. While aging does not modify the cascade condition, it slows down the cascade dynamics towards the full-adoption state: the exponential increase of adopters in time from the original model is replaced by a stretched exponential or power law, depending on the aging mechanism. Under several approximations, we give analytical expressions for the cascade condition and for the exponents of the adopters' density growth laws. Beyond random networks, we also describe by Monte Carlo simulations the effects of aging for the Threshold model in a two-dimensional lattice.

Vital node identification in hypergraphs via gravity model

Hypergraphs that can depict interactions beyond pairwise edges have emerged as an appropriate representation for modeling polyadic relations in complex systems. With the recent surge of interest in researching hypergraphs, the centrality problem has attracted much attention due to the challenge of how to utilize higher-order structure for the definition of centrality metrics. In this paper, we propose a new centrality method (HGC) on the basis of the gravity model as well as a semi-local HGC, which can achieve a balance between accuracy and computational complexity. Meanwhile, two comprehensive evaluation metrics, i.e., a complex contagion model in hypergraphs, which mimics the group influence during the spreading process and network s-efficiency based on the higher-order distance between nodes, are first proposed to evaluate the effectiveness of our methods. The results show that our methods can filter out nodes that have fast spreading ability and are vital in terms of hypergraph connectivity.

Class of models for random hypergraphs

Despite the recently exhibited importance of higher-order interactions for various processes, few flexible (null) models are available. In particular, most studies on hypergraphs focus on a small set of theoretical models. Here, we introduce a class of models for random hypergraphs which displays a similar level of flexibility of complex network models and where the main ingredient is the probability that a node belongs to a hyperedge. When this probability is a constant, we obtain a random hypergraph in the same spirit as the Erdos-Renyi graph. This framework also allows us to introduce different ingredients such as the preferential attachment for hypergraphs, or spatial random hypergraphs. In particular, we show that for the Erdos-Renyi case there is a transition threshold scaling as 1/sqrt[EN] where N is the number of nodes and E the number of hyperedges. We also discuss a random geometric hypergraph which displays a percolation transition for a threshold distance scaling as r_{c}^{*}∼1/sqrt[E]. For these various models, we provide results for the most interesting measures, and also introduce new ones in the spatial case for characterizing the geometrical properties of hyperedges. These different models might serve as benchmarks useful for analyzing empirical data.

Coupled spreading between information and epidemics on multiplex networks with simplicial complexes

The way of information diffusion among individuals can be quite complicated, and it is not only limited to one type of communication, but also impacted by multiple channels. Meanwhile, it is easier for an agent to accept an idea once the proportion of their friends who take it goes beyond a specific threshold. Furthermore, in social networks, some higher-order structures, such as simplicial complexes and hypergraph, can describe more abundant and realistic phenomena. Therefore, based on the classical multiplex network model coupling the infectious disease with its relevant information, we propose a novel epidemic model, in which the lower layer represents the physical contact network depicting the epidemic dissemination, while the upper layer stands for the online social network picturing the diffusion of information. In particular, the upper layer is generated by random simplicial complexes, among which the herd-like threshold model is adopted to characterize the information diffusion, and the unaware-aware-unaware model is also considered simultaneously. Using the microscopic Markov chain approach, we analyze the epidemic threshold of the proposed epidemic model and further check the results with numerous Monte Carlo simulations. It is discovered that the threshold model based on the random simplicial complexes network may still cause abrupt transitions on the epidemic threshold. It is also found that simplicial complexes may greatly influence the epidemic size at a steady state.

Opinion Dynamics with Higher-Order Bounded Confidence

The higher-order interactions in complex systems are gaining attention. Extending the classic bounded confidence model where an agent's opinion update is the average opinion of its peers, this paper proposes a higher-order version of the bounded confidence model. Each agent organizes a group opinion discussion among its peers. Then, the discussion's result influences all participants' opinions. Since an agent is also the peer of its peers, the agent actually participates in multiple group discussions. We assume the agent's opinion update is the average over multiple group discussions. The opinion dynamics rules can be arbitrary in each discussion. In this work, we experiment with two discussion rules: centralized and decentralized. We show that the centralized rule is equivalent to the classic bounded confidence model. The decentralized rule, however, can promote opinion consensus. In need of modeling specific real-life scenarios, the higher-order bounded confidence is more convenient to combine with other higher-order interactions, from the contagion process to evolutionary dynamics.

Two competing simplicial irreversible epidemics on simplicial complex

Higher-order interactions have significant implications for the dynamics of competing epidemic spreads. In this paper, a competing spread model for two simplicial irreversible epidemics (i.e., susceptible-infected-removed epidemics) on higher-order networks is proposed. The simplicial complexes are based on synthetic (including homogeneous and heterogeneous) and real-world networks. The spread process of two epidemics is theoretically analyzed by extending the microscopic Markov chain approach. When the two epidemics have the same 2-simplex infection rate and the 1-simplex infection rate of epidemic A ( _λ) is fixed at zero, an increase in the 1-simplex infection rate of epidemic B ( _λ) causes a transition from continuous growth to sharp growth in the spread of epidemic B with _λ. When _λ > 0, the growth of epidemic B is always continuous. With the increase of _λ, the outbreak threshold of epidemic B is delayed. When the difference in 1-simplex infection rates between the two epidemics reaches approximately three times, the stronger side obviously dominates. Otherwise, the coexistence of the two epidemics is always observed. When the 1-simplex infection rates are symmetrical, the increase in competition will accelerate the spread process and expand the spread area of both epidemics; when the 1-simplex infection rates are asymmetrical, the spread area of one epidemic increases with an increase in the 1-simplex infection rate from this epidemic while the other decreases. Finally, the influence of 2-simplex infection rates on the competing spread is discussed. An increase in 2-simplex infection rates leads to sharp growth in one of the epidemics.

Simplicial epidemic model with birth and death

In this paper, we propose a simplicial susceptible-infected-susceptible (SIS) epidemic model with birth and death to describe epidemic spreading based on group interactions, accompanying with birth and death. The site-based evolutions are formulated by the quenched mean-field probability equations for each site, which is a high-dimensional differential system. To facilitate a theoretical analysis of the influence of system parameters on dynamics, we adopt the mean-field method for our model to reduce the dimension. As a consequence, it suggests that birth and death rates influence the existence and stability of equilibria, as well as the appearance of a bistable state (the coexistence of the stable disease-free and endemic states), which is then confirmed by extensive simulations on empirical and synthetic networks. Furthermore, we find that another type of the bistable state in which a stable periodic outbreak state coexists with a steady disease-free state also emerges when birth and death rates and other parameters satisfy the certain conditions. Finally, we illustrate how the birth and death rates shift the density of infected nodes in the stationary state and the outbreak threshold, which is also verified by sensitivity analysis for the proposed model.

Disease extinction for susceptible-infected-susceptible models on dynamic graphs and hypergraphs

We consider stochastic, individual-level susceptible-infected-susceptible models for the spread of disease, opinion, or information on dynamic graphs and hypergraphs. We set up "snapshot" models where the interactions at any time are independently and identically sampled from an underlying distribution that represents a typical scenario. In the hypergraph case, this corresponds to a new Gilbert-style random hypergraph model. After justifying this modeling regime, we present useful mean field approximations. With an emphasis on the derivation of spectral conditions that determine long-time extinction, we give computational simulations and accompanying theoretical analysis for the exact models and their mean field approximations.

Echo chambers and information transmission biases in homophilic and heterophilic networks

We study how information transmission biases arise by the interplay between the structural properties of the network and the dynamics of the information in synthetic scale-free homophilic/heterophilic networks. We provide simple mathematical tools to quantify these biases. Both Simple and Complex Contagion models are insufficient to predict significant biases. In contrast, a Hybrid Contagion model—in which both Simple and Complex Contagion occur—gives rise to three different homophily-dependent biases: emissivity and receptivity biases, and echo chambers. Simulations in an empirical network with high homophily confirm our findings. Our results shed light on the mechanisms that cause inequalities in the visibility of information sources, reduced access to information, and lack of communication among distinct groups.

Full reconstruction of simplicial complexes from binary contagion and Ising data

Previous efforts on data-based reconstruction focused on complex networks with pairwise or two-body interactions. There is a growing interest in networks with higher-order or many-body interactions, raising the need to reconstruct such networks based on observational data. We develop a general framework combining statistical inference and expectation maximization to fully reconstruct 2-simplicial complexes with two- and three-body interactions based on binary time-series data from two types of discrete-state dynamics. We further articulate a two-step scheme to improve the reconstruction accuracy while significantly reducing the computational load. Through synthetic and real-world 2-simplicial complexes, we validate the framework by showing that all the connections can be faithfully identified and the full topology of the 2-simplicial complexes can be inferred. The effects of noisy data or stochastic disturbance are studied, demonstrating the robustness of the proposed framework.

Data-driven recovery of topology is challenging for networks beyond pairwise interactions. The authors propose a framework to reconstruct complex networks with higher-order interactions from time series, focusing on networks with 2-simplexes where social contagion and Ising dynamics generate binary data.

Hypergraph assortativity: A dynamical systems perspective

The largest eigenvalue of the matrix describing a network's contact structure is often important in predicting the behavior of dynamical processes. We extend this notion to hypergraphs and motivate the importance of an analogous eigenvalue, the expansion eigenvalue, for hypergraph dynamical processes. Using a mean-field approach, we derive an approximation to the expansion eigenvalue in terms of the degree sequence for uncorrelated hypergraphs. We introduce a generative model for hypergraphs that includes degree assortativity, and use a perturbation approach to derive an approximation to the expansion eigenvalue for assortative hypergraphs. We define the dynamical assortativity, a dynamically sensible definition of assortativity for uniform hypergraphs, and describe how reducing the dynamical assortativity of hypergraphs through preferential rewiring can extinguish epidemics. We validate our results with both synthetic and empirical datasets.

Dynamics on higher-order networks: a review

Network science has evolved into an indispensable platform for studying complex systems. But recent research has identified limits of classical networks, where links connect pairs of nodes, to comprehensively describe group interactions. Higher-order networks, where a link can connect more than two nodes, have therefore emerged as a new frontier in network science. Since group interactions are common in social, biological and technological systems, higher-order networks have recently led to important new discoveries across many fields of research. Here, we review these works, focusing in particular on the novel aspects of the dynamics that emerges on higher-order networks. We cover a variety of dynamical processes that have thus far been studied, including different synchronization phenomena, contagion processes, the evolution of cooperation and consensus formation. We also outline open challenges and promising directions for future research.

A system model of three-body interactions in complex networks: consensus and conservation

Networked complex systems in a wide range of physics, biology and social sciences involve synergy among multiple agents beyond pairwise interactions. Higher-order mathematical structures such as hypergraphs have been increasingly popular in modelling and analysis of complex dynamical behaviours. Here, we study a simple three-body consensus model, which favourably incorporates higher-order network interactions, higher-order dimensional states, the group reinforcement effect and the social homophily principle. The model features asymmetric roles of acting agents using modulating functions. We analytically establish sufficient conditions for nonlinear consensus and conservation of states for agents with both discrete-time and continuous-time dynamics. We show that higher-order interactions encoded in three-body edges give rise to consensus and conservation for systems with gravity-like and Heaviside-like modulating functions. Furthermore, we illustrate our theoretical results with numerical simulations and examine the system convergence time through a network depreciation process.

Dynamics of the threshold model on hypergraphs

The threshold model has been widely adopted as a prototype for studying contagion processes on social networks. In this paper, we consider individual interactions in groups of three or more vertices and study the threshold model on hypergraphs. To understand how high-order interactions affect the breakdown of the system, we develop a theoretical framework based on generating function technology to derive the cascade condition and the giant component of vulnerable vertices, which depend on both hyperedges and hyperdegrees. First, we find a dual role of the hyperedge in propagation: when the average hyperdegree is small, increasing the size of the hyperedges may make the system fragile, while the average hyperdegree is relatively large, the increase of the hyperedges causes the system to be robust. Then, we identify the effects of threshold, hyperdegree, and hyperedge heterogeneities. The heterogeneity of individual thresholds causes the system to be more fragile, while the heterogeneity of individual hyperdegrees or hyperedges increases the robustness of the system. Finally, we show that the higher hyperdegree a vertex has, the larger possibility and faster speed it will get activated. We verify these results by simulating meme spreading on both random hypergraph models and hypergraphs constructed from empirical data.

Topological analysis of the latent geometry of a complex network

Most real-world networks are embedded in latent geometries. If a node in a network is found in the vicinity of another node in the latent geometry, the two nodes have a disproportionately high probability of being connected by a link. The latent geometry of a complex network is a central topic of research in network science, which has an expansive range of practical applications, such as efficient navigation, missing link prediction, and brain mapping. Despite the important role of topology in the structures and functions of complex systems, little to no study has been conducted to develop a method to estimate the general unknown latent geometry of complex networks. Topological data analysis, which has attracted extensive attention in the research community owing to its convincing performance, can be directly implemented into complex networks; however, even a small fraction (0.1%) of long-range links can completely erase the topological signature of the latent geometry. Inspired by the fact that long-range links in a network have disproportionately high loads, we develop a set of methods that can analyze the latent geometry of a complex network: the modified persistent homology diagram and the map of the latent geometry. These methods successfully reveal the topological properties of the synthetic and empirical networks used to validate the proposed methods.

Local topological moves determine global diffusion properties of hyperbolic higher-order networks

From social interactions to the human brain, higher-order networks are key to describe the underlying network geometry and topology of many complex systems. While it is well known that network structure strongly affects its function, the role that network topology and geometry has on the emerging dynamical properties of higher-order networks is yet to be clarified. In this perspective, the spectral dimension plays a key role since it determines the effective dimension for diffusion processes on a network. Despite its relevance, a theoretical understanding of which mechanisms lead to a finite spectral dimension, and how this can be controlled, still represents a challenge and is the object of intense research. Here, we introduce two nonequilibrium models of hyperbolic higher-order networks and we characterize their network topology and geometry by investigating the intertwined appearance of small-world behavior, δ-hyperbolicity, and community structure. We show that different topological moves, determining the nonequilibrium growth of the higher-order hyperbolic network models, induce tuneable values of the spectral dimension, showing a rich phenomenology which is not displayed in random graph ensembles. In particular, we observe that, if the topological moves used to construct the higher-order network increase the area/volume ratio, then the spectral dimension continuously decreases, while the opposite effect is observed if the topological moves decrease the area/volume ratio. Our work reveals a new link between the geometry of a network and its diffusion properties, contributing to a better understanding of the complex interplay between network structure and dynamics.

Higher-order percolation processes on multiplex hypergraphs

Higher-order interactions are increasingly recognized as a fundamental aspect of complex systems ranging from the brain to social contact networks. Hypergraphs as well as simplicial complexes capture the higher-order interactions of complex systems and allow us to investigate the relation between their higher-order structure and their function. Here we establish a general framework for assessing hypergraph robustness and we characterize the critical properties of simple and higher-order percolation processes. This general framework builds on the formulation of the random multiplex hypergraph ensemble where each layer is characterized by hyperedges of given cardinality. We observe that in presence of the structural cutoff the ensemble of multiplex hypergraphs can be mapped to an ensemble of multiplex bipartite networks. We reveal the relation between higher-order percolation processes in random multiplex hypergraphs, interdependent percolation of multiplex networks, and K-core percolation. The structural correlations of the random multiplex hypergraphs are shown to have a significant effect on their percolation properties. The wide range of critical behaviors observed for higher-order percolation processes on multiplex hypergraphs elucidates the mechanisms responsible for the emergence of discontinuous transition and uncovers interesting critical properties which can be applied to the study of epidemic spreading and contagion processes on higher-order networks.

Universal Nonlinear Infection Kernel from Heterogeneous Exposure on Higher-Order Networks

The collocation of individuals in different environments is an important prerequisite for exposure to infectious diseases on a social network. Standard epidemic models fail to capture the potential complexity of this scenario by (1) neglecting the higher-order structure of contacts that typically occur through environments like workplaces, restaurants, and households, and (2) assuming a linear relationship between the exposure to infected contacts and the risk of infection. Here, we leverage a hypergraph model to embrace the heterogeneity of environments and the heterogeneity of individual participation in these environments. We find that combining heterogeneous exposure with the concept of minimal infective dose induces a universal nonlinear relationship between infected contacts and infection risk. Under nonlinear infection kernels, conventional epidemic wisdom breaks down with the emergence of discontinuous transitions, superexponential spread, and hysteresis.

Epidemics on hypergraphs: spectral thresholds for extinction

Epidemic spreading is well understood when a disease propagates around a contact graph. In a stochastic susceptible–infected–susceptible setting, spectral conditions characterize whether the disease vanishes. However, modelling human interactions using a graph is a simplification which only considers pairwise relationships. This does not fully represent the more realistic case where people meet in groups. Hyperedges can be used to record higher order interactions, yielding more faithful and flexible models and allowing for the rate of infection of a node to depend on group size and also to vary as a nonlinear function of the number of infectious neighbours. We discuss different types of contagion models in this hypergraph setting and derive spectral conditions that characterize whether the disease vanishes. We study both the exact individual-level stochastic model and a deterministic mean field ODE approximation. Numerical simulations are provided to illustrate the analysis. We also interpret our results and show how the hypergraph model allows us to distinguish between contributions to infectiousness that (i) are inherent in the nature of the pathogen and (ii) arise from behavioural choices (such as social distancing, increased hygiene and use of masks). This raises the possibility of more accurately quantifying the effect of interventions that are designed to contain the spread of a virus.

Generative hypergraph clustering: From blockmodels to modularity

Hypergraphs are a natural modeling paradigm for networked systems with multiway interactions. A standard task in network analysis is the identification of closely related or densely interconnected nodes. We propose a probabilistic generative model of clustered hypergraphs with heterogeneous node degrees and edge sizes. Approximate maximum likelihood inference in this model leads to a clustering objective that generalizes the popular modularity objective for graphs. From this, we derive an inference algorithm that generalizes the Louvain graph community detection method, and a faster, specialized variant in which edges are expected to lie fully within clusters. Using synthetic and empirical data, we demonstrate that the specialized method is highly scalable and can detect clusters where graph-based methods fail. We also use our model to find interpretable higher-order structure in school contact networks, U.S. congressional bill cosponsorship and committees, product categories in copurchasing behavior, and hotel locations from web browsing sessions.

Betweenness centrality of teams in social networks

Betweenness centrality (BC) was proposed as an indicator of the extent of an individual's influence in a social network. It is measured by counting how many times a vertex (i.e., an individual) appears on all the shortest paths between pairs of vertices. A question naturally arises as to how the influence of a team or group in a social network can be measured. Here, we propose a method of measuring this influence on a bipartite graph comprising vertices (individuals) and hyperedges (teams). When the hyperedge size varies, the number of shortest paths between two vertices in a hypergraph can be larger than that in a binary graph. Thus, the power-law behavior of the team BC distribution breaks down in scale-free hypergraphs. However, when the weight of each hyperedge, for example, the performance per team member, is counted, the team BC distribution is found to exhibit power-law behavior. We find that a team with a widely connected member is highly influential.

Discordant synchronization patterns on directed networks of identical phase oscillators with attractive and repulsive couplings

We study the collective dynamics of identical phase oscillators on globally coupled networks whose interactions are asymmetric and mediated by positive and negative couplings. We split the set of oscillators into two interconnected subpopulations. In this setup, oscillators belonging to the same group interact via symmetric couplings while the interaction between subpopulations occurs in an asymmetric fashion. By employing the dimensional reduction scheme of the Ott-Antonsen (OA) theory, we verify the existence of traveling wave and π-states, in addition to the classical fully synchronized and incoherent states. Bistability between all collective states is reported. Analytical results are generally in excellent agreement with simulations; for some parameters and initial conditions, however, we numerically detect chimera-like states which are not captured by the OA theory.

Diffusion geometry of multiplex and interdependent systems

Complex networks are characterized by latent geometries induced by their topology or by the dynamics on them. In the latter case, different network-driven processes induce distinct geometric features that can be captured by adequate metrics. Random walks, a proxy for a broad spectrum of processes, from simple contagion to metastable synchronization and consensus, have been recently used, Domenico [Phys. Rev. Lett. 118, 168301 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.168301] to define the class of diffusion geometries and pinpoint the functional mesoscale organization of complex networks from a genuine geometric perspective. Here we first extend this class to families of distinct random walk dynamics-including local and nonlocal information-on multilayer networks-a paradigm for biological, neural, social, transportation, and financial systems-overcoming limitations such as the presence of isolated nodes and disconnected components, typical of real-world networks. We then characterize the multilayer diffusion geometry of synthetic and empirical systems, highlighting the role played by different random search dynamics in shaping the geometric features of the corresponding diffusion manifolds.

Simplicial SIRS epidemic models with nonlinear incidence rates

Mathematical epidemiology that describes the complex dynamics on social networks has become increasingly popular. However, a few methods have tackled the problem of coupling network topology with complex incidence mechanisms. Here, we propose a simplicial susceptible-infected-recovered-susceptible (SIRS) model to investigate the epidemic spreading via combining the network higher-order structure with a nonlinear incidence rate. A network-based social system is reshaped to a simplicial complex, in which the spreading or infection occurs with nonlinear reinforcement characterized by the simplex dimensions. Compared with the previous simplicial susceptible-infected-susceptible (SIS) models, the proposed SIRS model can not only capture the discontinuous transition and the bistability of a complex system but also capture the periodic phenomenon of epidemic outbreaks. More significantly, the two thresholds associated with the bistable region and the critical value of the reinforcement factor are derived. We further analyze the stability of equilibrium points of the proposed model and obtain the condition of existence of the bistable states and limit cycles. This work expands the simplicial SIS models to SIRS models and sheds light on a novel perspective of combining the higher-order structure of complex systems with nonlinear incidence rates.

Homological percolation transitions in growing simplicial complexes

Simplicial complex (SC) representation is an elegant mathematical framework for representing the effect of complexes or groups with higher-order interactions in a variety of complex systems ranging from brain networks to social relationships. Here, we explore the homological percolation transitions (HPTs) of growing SCs using empirical datasets and model studies. The HPTs are determined by the first and second Betti numbers, which indicate the appearance of one- and two-dimensional macroscopic-scale homological cycles and cavities, respectively. A minimal SC model with two essential factors, namely, growth and preferential attachment, is proposed to model social coauthorship relationships. This model successfully reproduces the HPTs and determines the transition types as an infinite-order Berezinskii-Kosterlitz-Thouless type but with different critical exponents. In contrast to the Kahle localization observed in static random SCs, the first Betti number continues to increase even after the second Betti number appears. This delocalization is found to stem from the two aforementioned factors and arises when the merging rate of two-dimensional simplexes is less than the birth rate of isolated simplexes. Our results can provide a topological insight into the maturing steps of complex networks such as social and biological networks.

Temporal properties of higher-order interactions in social networks

Human social interactions in local settings can be experimentally detected by recording the physical proximity and orientation of people. Such interactions, approximating face-to-face communications, can be effectively represented as time varying social networks with links being unceasingly created and destroyed over time. Traditional analyses of temporal networks have addressed mostly pairwise interactions, where links describe dyadic connections among individuals. However, many network dynamics are hardly ascribable to pairwise settings but often comprise larger groups, which are better described by higher-order interactions. Here we investigate the higher-order organizations of temporal social networks by analyzing five publicly available datasets collected in different social settings. We find that higher-order interactions are ubiquitous and, similarly to their pairwise counterparts, characterized by heterogeneous dynamics, with bursty trains of rapidly recurring higher-order events separated by long periods of inactivity. We investigate the evolution and formation of groups by looking at the transition rates between different higher-order structures. We find that in more spontaneous social settings, group are characterized by slower formation and disaggregation, while in work settings these phenomena are more abrupt, possibly reflecting pre-organized social dynamics. Finally, we observe temporal reinforcement suggesting that the longer a group stays together the higher the probability that the same interaction pattern persist in the future. Our findings suggest the importance of considering the higher-order structure of social interactions when investigating human temporal dynamics.

The aging effect in evolving scientific citation networks

The study of citation networks is of interest to the scientific community. However, the underlying mechanism driving individual citation behavior remains imperfectly understood, despite the recent proliferation of quantitative research methods. Traditional network models normally use graph theory to consider articles as nodes and citations as pairwise relationships between them. In this paper, we propose an alternative evolutionary model based on hypergraph theory in which one hyperedge can have an arbitrary number of nodes, combined with an aging effect to reflect the temporal dynamics of scientific citation behavior. Both theoretical approximate solution and simulation analysis of the model are developed and validated using two benchmark datasets from different disciplines, i.e. publications of the American Physical Society (APS) and the Digital Bibliography & Library Project (DBLP). Further analysis indicates that the attraction of early publications will decay exponentially. Moreover, the experimental results show that the aging effect indeed has a significant influence on the description of collective citation patterns. Shedding light on the complex dynamics driving these mechanisms facilitates the understanding of the laws governing scientific evolution and the quantitative evaluation of scientific outputs.

Master equation analysis of mesoscopic localization in contagion dynamics on higher-order networks

Simple models of infectious diseases tend to assume random mixing of individuals, but real interactions are not random pairwise encounters: they occur within various types of gatherings such as workplaces, households, schools, and concerts, best described by a higher-order network structure. We model contagions on higher-order networks using group-based approximate master equations, in which we track all states and interactions within a group of nodes and assume a mean-field coupling between them. Using the susceptible-infected-susceptible dynamics, our approach reveals the existence of a mesoscopic localization regime, where a disease can concentrate and self-sustain only around large groups in the network overall organization. In this regime, the phase transition is smeared, characterized by an inhomogeneous activation of the groups. At the mesoscopic level, we observe that the distribution of infected nodes within groups of the same size can be very dispersed, even bimodal. When considering heterogeneous networks, both at the level of nodes and at the level of groups, we characterize analytically the region associated with mesoscopic localization in the structural parameter space. We put in perspective this phenomenon with eigenvector localization and discuss how a focus on higher-order structures is needed to discern the more subtle localization at the mesoscopic level. Finally, we discuss how mesoscopic localization affects the response to structural interventions and how this framework could provide important insights for a broad range of dynamics.

The Basic Reproduction Number as a Loop Gain Matrix

The COVID-19 pandemic and the disordered reactions of most governments made the importance of mathematical modelling and model-based predictions evident, even outside the scientific community. The basic reproduction number [Formula: see text] quickly entered the common jargon, as a concise but effective tool to communicate the spreading power of a disease and estimate, at least roughly, the possible outcomes of the epidemic. However, while [Formula: see text] is easily defined for simple models, its proper definition is more subtle for larger, state-of-the-art models. Here we show that it is nothing else than the spectral radius of the gain matrix of a linear system, and that this matrix generalizes [Formula: see text] in the computation of the vector-valued final epidemic size and epidemic threshold, in a large class of finite-dimensional SIR-like models.