Stability of synchronization in simplicial complexes

State estimation in water distribution system via diffusion on the edge space

The steady state of a water distribution system abides by the laws of mass and energy conservation. Hydraulic solvers, such as the one used by EPANET approach the simulation for a given topology with a Newton-Raphson algorithm. However, iterative approximation involves a matrix inversion which acts as a computational bottleneck and may significantly slow down the process. In this work, we propose to rethink the current approach for steady state estimation to leverage the recent advancements in Graphics Processing Unit (GPU) hardware. Modern GPUs enhance matrix multiplication and enable memory-efficient sparse matrix operations, allowing for massive parallelization. Such features are particularly beneficial for state estimation in infrastructure networks, which are characterized by sparse connectivity between system elements. To realize this approach and tap into the potential of GPU-enhanced parallelization, we reformulate the problem as a diffusion process on the edges of a graph. Edge-based diffusion is inherently related to conservation laws governing a water distribution system. Using a numerical approximation scheme, the diffusion leads to a state of the system that satisfies mass and energy conservation principles. Using existing benchmark water distribution systems, we show that the proposed method allows parallelizing thousands of hydraulic simulations simultaneously with very high accuracy.

Hypergraph reconstruction from dynamics

A plethora of methods have been developed in the past two decades to infer the underlying network structure of an interconnected system from its collective dynamics. However, methods capable of inferring nonpairwise interactions are only starting to appear. Here, we develop an inference algorithm based on sparse identification of nonlinear dynamics (SINDy) to reconstruct hypergraphs and simplicial complexes from time-series data. Our model-free method does not require information about node dynamics or coupling functions, making it applicable to complex systems that do not have a reliable mathematical description. We first benchmark the new method on synthetic data generated from Kuramoto and Lorenz dynamics. We then use it to infer the effective connectivity in the brain from resting-state EEG data, which reveals significant contributions from non-pairwise interactions in shaping the macroscopic brain dynamics.

Bifurcations and collective states of Kuramoto oscillators with higher-order interactions and rotational symmetry breaking

We study a network of identical Kuramoto oscillators with higher-order interactions that also break the rotational symmetry of the system. To gain analytical insights into this model, we use the Watanabe-Strogatz Ansatz, which allows us to reduce the dimensionality of the original system of equations. The study of stability and bifurcations of the reduced system reveals a codimension two Bogdanov-Takens bifurcation and several other associated bifurcations. Such analysis is corroborated by numerical simulations of the associated Kuramoto system, which, in turn, unveils a variety of collective behaviors such as synchronized motion, oscillation death, chimeras, incoherent states, and traveling waves. Importantly, this system displays a case where alternating chimeras emerge in an indistinguishable single population of oscillators, which may offer insights into the unihemispheric slow-wave sleep phenomenon observed in mammals and birds.

Hyperedge overlap drives synchronizability of systems with higher-order interactions

The microscopic organization of dynamical systems coupled via higher-order interactions plays a pivotal role in understanding their collective behavior. In this paper, we introduce a framework for systematically investigating the impact of the interaction structure on dynamical processes. Specifically, we develop an hyperedge overlap matrix whose elements characterize the two main aspects of the microscopic organization of higher-order interactions: the inter-order hyperedge overlap (nondiagonal matrix elements) and the intra-order hyperedge overlap (encapsulated in the diagonal elements). In this way, the first set of terms quantifies the extent of superposition of nodes among hyperedges of different orders, while the second focuses on the number of nodes in common between hyperedges of the same order. Our findings indicate that large values of both types of hyperedge overlap hinder synchronization stability, and that the larger is the order of interactions involved, the more important is their role. Our findings also indicate that the two types of overlap have qualitatively distinct effects on the dynamics of coupled chaotic oscillators. In particular, large values of intra-order hyperedge overlap hamper synchronization by favoring the presence of disconnected sets of hyperedges, while large values of inter-order hyperedge overlap hinder synchronization by increasing the number of shared nodes between groups converging on different trajectories, without necessarily causing disconnected sets of hyperedges.

Study on the robust control of higher-order networks

With the development of information technology, the interactions between nodes are no longer restricted to two nodes. Recently, researchers have proposed a higher-order network, which is more suitable to describe the multidimensional interaction relationships in systems. A higher-order network with good robustness can effectively resist natural disasters and deliberate attacks. How to improve the robustness of the higher-order network is worth studying. In this paper, we construct two higher-order networks based on the simplex structure. In addition, we propose a capacity load model that can describe the robustness of higher-order networks. The simulation results show that the robustness of the higher-order network is positively correlated with the size of the high-order network, the larger the size of the higher-order network, the more robust the higher-order network is in two attack strategies. In addition, the robustness of higher-order is related to the number of 2-simplexes in the network. Furthermore, the robustness is affected by the weight coefficients of 1-simplex and 2-simplex interactions. Therefore, we can improve robustness of higher-order networks by controlling the weight coefficients of the 1- and 2-simplex in higher-order networks. We verified the conclusions by two synthetic higher-order networks and a constructed higher-order network based on real data.

The triangular structure beyond pairwise interactions affects the robustness of the world trade networks

Unlike hollow triangles formed through pairwise interactions, a filled triangle or two-simplex comprises three nodes that form a group and represent the most fundamental higher-order interaction. To analyze the effects of higher-order triangles on the robustness of world trade networks, we integrate multilateral regional trade agreements and import-export world trade data to construct two-simplex higher-order trade networks. The topological characteristics indicate a significant growth in the scale and complexity of trade networks over time, with a notable decline in 2020. Then, we introduce node attack strategies designed to simulate scenarios where the key countries or regions withdraw from the trade network. It is revealed that network robustness has improved along with size and complexity, although it diminished in 2020. To further explore the factors influencing the changes in network robustness, we generate higher-order synthetic trade networks based on the random simplicial complex (RSC) model and the scale-free simplicial complex (SFSC) model. The synthetic trade networks demonstrate that increasing the average degree enhances robustness, while merely increasing the number of nodes or filled triangles can weaken it. Additionally, scale-free higher-order networks exhibit lower robustness due to vulnerability of the hub nodes, in contrast to the higher resilience of random simplicial complexes. These insights emphasize the importance of fostering multilateral interactions and strengthening ties for network robustness.

Spatiotemporal Complexity in the Psychotic Brain

Psychotic disorders, such as schizophrenia and bipolar disorder, pose significant diagnostic challenges with major implications on mental health. The measures of resting-state fMRI spatiotemporal complexity offer a powerful tool for identifying irregularities in brain activity. To capture global brain connectivity, we employed information-theoretic metrics, overcoming the limitations of pairwise correlation analysis approaches. This enables a more comprehensive exploration of higher-order interactions and multiscale intrinsic connectivity networks (ICNs) in the psychotic brain. In this study, we provide converging evidence suggesting that the psychotic brain exhibits states of randomness across both spatial and temporal dimensions. To further investigate these disruptions, we estimated brain network connectivity using redundancy and synergy measures, aiming to assess the integration and segregation of topological information in the psychotic brain. Our findings reveal a disruption in the balance between redundant and synergistic information, a phenomenon we term brainquake in this study, which highlights the instability and disorganization of brain networks in psychosis. Moreover, our exploration of higher-order topological functional connectivity reveals profound disruptions in brain information integration. Aberrant information interactions were observed across both cortical and subcortical ICNs. We specifically identified the most easily affected irregularities in the sensorimotor, visual, temporal, default mode, and fronto-parietal networks, as well as in the hippocampal and amygdalar regions, all of which showed disruptions. These findings underscore the severe impact of psychotic states on multiscale critical brain networks, suggesting a profound alteration in the brain's complexity and organizational states.

Hyperedge overlap drives explosive transitions in systems with higher-order interactions

Recent studies have shown that novel collective behaviors emerge in complex systems due to the presence of higher-order interactions. However, how the collective behavior of a system is influenced by the microscopic organization of its higher-order interactions is not fully understood. In this work, we introduce a way to quantify the overlap among the hyperedges of a higher-order network, and we show that real-world systems exhibit different levels of intra-order hyperedge overlap. We then study two types of dynamical processes on higher-order networks, namely complex contagion and synchronization, finding that intra-order hyperedge overlap plays a universal role in determining the collective behavior in a variety of systems. Our results demonstrate that the presence of higher-order interactions alone does not guarantee abrupt transitions. Rather, explosivity and bistability require a microscopic organization of the structure with a low value of intra-order hyperedge overlap.

Synchronization stability of epileptic brain network with higher-order interactions

Generally, epilepsy is considered as abnormally enhanced neuronal excitability and synchronization. So far, previous studies on the synchronization of epileptic brain networks mainly focused on the synchronization strength, but the synchronization stability has not yet been explored as deserved. In this paper, we propose a novel idea to construct a hypergraph brain network (HGBN) based on phase synchronization. Furthermore, we apply the synchronization stability framework of the nonlinear coupled oscillation dynamic model (generalized Kuramoto model) to investigate the HGBNs of epilepsy patients. Specifically, the synchronization stability of the epileptic brain is quantified by calculating the eigenvalue spectrum of the higher-order Laplacian matrix in HGBN. Results show that synchronization stability decreased slightly in the early stages of seizure but increased significantly prior to seizure termination. This indicates that an emergency self-regulation mechanism of the brain may facilitate the termination of seizures. Moreover, the variation in synchronization stability during epileptic seizures may be induced by the topological changes of epileptogenic zones (EZs) in HGBN. Finally, we verify that the higher-order interactions improve the synchronization stability of HGBN. This study proves the validity of the synchronization stability framework with the nonlinear coupled oscillation dynamical model in HGBN, emphasizing the importance of higher-order interactions and the influence of EZs on the termination of epileptic seizures.

Experimental datasets on synchronization in simplicial complexes

Some real-world phenomena and human-made problems have been modeled as networks where the objects form pairwise interactions. However, this is a limited approach when the existence of high-order interactions is inherent in a system, such as the brain, social networks and ecosystems. The way in which these high-order interactions affect the collective behavior of a complex system is still an open question. For this reason, it is necessary to analyze theoretically, numerically and experimentally the consequences of higher-order interactions in complex systems. Here, we provide experimental datasets of the dynamics of three nonlinear electronic oscillators, namely, Rössler oscillators, interacting into a simplicial complex whose connections rely on both linear (diffusive) and nonlinear (high-order) coupling. It is well-known that Rössler systems only achieve the synchronization when they are coupled by means of x or y variable. Considering this fact, we designed our experiment considering four scenarios. The first one, when both linear and nonlinear coupling functions are introduced through the x variable. The second one, occurring when linear coupling is introduced through the x variable and the nonlinear coupling through the y variable. The third case happens when the linear coupling is introduced through the y variable whereas nonlinear coupling goes through the x variable. The last case, when both linear and nonlinear coupling are introduced through the y variable. For each scenario, we acquired 10000 times series when both the linear and nonlinear coupling strengths were modified. Each time series contained 30000 temporal points. These datasets are useful to corroborate the conditions to reach the synchronized state varying the linear/non-linear coupling strengths and to test new metrics for better understanding the effects of higher-order interactions in complex networks.

Developing the 'omic toolkit of comparative physiologists

Typical 'omic analyses reduce complex biological systems to simple lists of supposedly independent variables, failing to account for changes in the wider transcriptional landscape. In this commentary, we discuss the utility of network approaches for incorporating this wider context into the study of physiological phenomena. We highlight opportunities to build on traditional network tools by utilising cutting-edge techniques to account for higher order interactions (i.e. beyond pairwise associations) within datasets, allowing for more accurate models of complex 'omic systems. Finally, we show examples of previous works utilising network approaches to gain additional insight into their organisms of interest. As 'omics grow in both their popularity and breadth of application, so does the requirement for flexible analytical tools capable of interpreting and synthesising complex datasets.

Stability of synchronization manifolds and its nonlinear behaviour in memristive coupled discrete neuron model

In this study, we investigate the impact of first and second-order coupling strengths on the stability of a synchronization manifold in a Discrete FitzHugh-Nagumo (DFHN) neuron model with memristor coupling. Master Stability Function (MSF) is used to estimate the stability of the synchronized manifold. The MSF of the DFHN model exhibits two zero crossings as we vary the coupling strengths, which is categorized as class Γ 2 . Interestingly, both zero-crossing points demonstrate a power-law relationship with respect to both the first-order coupling strength and flux coefficient, as well as the second-order coupling strength and flux coefficient. In contrast, the zero crossings follow a linear relationship between first-order and second-order coupling strength. These linear and nonlinear relationships enable us to forecast the zero-crossing point and, consequently, determine the coupling strengths at which the stability of the synchronization manifold changes for any given set of parameters. We further explore the regime of the stable synchronization manifold within a defined parameter space. Lower values of both first and second-order coupling strengths have minimal impact on the transition between stable and unstable synchronization regimes. Conversely, higher coupling strengths lead to a shrinking regime of the stable synchronization manifold. This reduction follows an exponential relationship with the coupling strengths. This study is helpful in brain-inspired computing systems by understanding synchronization stability in neuron models with memristor coupling. It helps to create more efficient neural networks for tasks like pattern recognition and data processing.

How synchronized human networks escape local minima

Finding the global minimum in complex networks while avoiding local minima is challenging in many types of networks. In human networks and communities, adapting and finding new stable states amid changing conditions due to conflicts, climate changes, or disasters, is crucial. We studied the dynamics of complex networks of violin players and observed that such human networks have different methods to avoid local minima than other non-human networks. Humans can change the coupling strength between them or change their tempo. This leads to different dynamics than other networks and makes human networks more robust and better resilient against perturbations. We observed high-order vortex states, oscillation death, and amplitude death, due to the unique dynamics of the network. This research may have implications in politics, economics, pandemic control, decision-making, and predicting the dynamics of networks with artificial intelligence.

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.

Pattern dynamics of networked epidemic model with higher-order infections

Current research on pattern formations in networked reaction-diffusion (RD) systems predominantly focuses on the impacts of diffusion heterogeneity between nodes, often overlooking the contact heterogeneity between individuals within nodes in the reaction terms. In this paper, we establish a networked RD model incorporating infection through higher-order interaction in simplicial complexes in the reaction terms. Through theoretical and numerical analysis, we find that these higher-order interactions may induce Turing instability in the system. Notably, the relationship between the size of the Turing instability range and the average 2-simplices degree within nodes can be approximated by a quadratic function. Additionally, as the average 2-simplices degree increases, the amplitude of the patterns exhibits three distinct trends: increasing, decreasing, and initially increasing then decreasing, while the average infection density increases consistently. We then provide a possible explanation for these observations. Our findings offer new insights into the effects of contact heterogeneity within nodes on networked pattern formations, thereby informing the development of epidemic prevention and control measures.

Chimera-inspired dynamics: When higher-order interactions are expressed differently

The exploration of chimera-inspired dynamics in nonlocally coupled networks of Kuramoto oscillators with higher-order interactions is still in its nascent stages. Concurrently, the investigation of collective phenomena in higher-order interaction networks is gaining attraction. Here, we observe that hypergraph networks tend to synchronize through lower-order interactions, whereas simplicial complex networks exhibit a preference for higher-order interactions. This observation suggests that higher-order representations manifest substantial differences in chimera-inspired synchronization regions. Moreover, we introduce an explicit expression for identifying the chimera state. With a comprehensive basin stability analysis and the interplay of pairwise and higher-order interaction strengths, the emergence of the chimera state is inherent in high-order interaction networks. Our findings contribute to the understanding of chimera-inspired dynamics in higher-order interaction networks.

Coexistence of multistable synchronous states in a three-oscillator system with higher-order interaction

We study a three-oscillator system with pairwise (1-simplex) and triadic (2-simplex) interactions, and focus on how the interplay between these two types of interactions influences synchronous dynamics. Using a minimal model, dynamical phenomena in systems that have been previously studied under the thermodynamic limit (N→∞) are further clarified. Various synchronous states, including in-phase and antiphase synchronous states, as well as partial synchronous states are demonstrated. Meanwhile, significant multistable behaviors are revealed. Our work extends previous research on pairwise and triadic interactions, which can deepen our understanding of the impact of correlation between higher-order interaction and multistability. These dynamic phenomena bear resemblance to the diverse synchronization patterns of the heart, and they also serve as pivotal factors in information storage and memory retention within the brain.

Effects of high-order interactions on synchronization of a fractional-order neural system

In this study, effects of high-order interactions on synchronization of the fractional-order Hindmarsh-Rose neuron models have been examined deeply. Three different network situations in which first-order coupling, high-order couplings and first-plus second-order couplings included in the neuron models, have been considered, respectively. In order to find the optimal values of the first- and high-order coupling parameters by minimizing the cost function resulted from pairwise and triple interactions, the particle swarm optimization algorithm is employed. It has been deduced from the numerical simulation results that the first-plus second-order couplings induce the synchronization with both reduced first-order coupling strength and total cost compared to the first-order coupled case solely. When the only first-order coupled case is compared with the only second-order coupled case, it is determined that the neural network with only second-order couplings involved could achieve synchronization with lower coupling strength and, as a natural result, lower cost. On the other hand, solely second- and first-plus second-order coupled networks give very similar results each other. Therefore, high-order interactions have a positive effect on the synchronization. Additionally, increasing the network size decreases the values of the both first- and high-order coupling strengths to reach synchronization. However, in this case, total cost should be kept in the mind. Decreasing the fractional order parameter causes slower synchronization due to the decreased frequency of the neural response. On the other hand, more synchronous network is possible with increasing the fractional order parameter. Thus, the neural network with higher fractional order as well as high-order coupled is a good candidate in terms of the neural synchronization.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11571-023-10055-z.

On the higher-order smallest ring-star network of Chialvo neurons under diffusive couplings

Network dynamical systems with higher-order interactions are a current trending topic, pervasive in many applied fields. However, our focus in this work is neurodynamics. We numerically study the dynamics of the smallest higher-order network of neurons arranged in a ring-star topology. The dynamics of each node in this network is governed by the Chialvo neuron map, and they interact via linear diffusive couplings. This model is perceived to imitate the nonlinear dynamical properties exhibited by a realistic nervous system where the neurons transfer information through multi-body interactions. We deploy the higher-order coupling strength as the primary bifurcation parameter. We start by analyzing our model using standard tools from dynamical systems theory: fixed point analysis, Jacobian matrix, and bifurcation patterns. We observe the coexistence of disparate chaotic attractors. We also observe an interesting route to chaos from a fixed point via period-doubling and the appearance of cyclic quasiperiodic closed invariant curves. Furthermore, we numerically observe the existence of codimension-1 bifurcation points: saddle-node, period-doubling, and Neimark-Sacker. We also qualitatively study the typical phase portraits of the system, and numerically quantify chaos and complexity using the 0-1 test and sample entropy measure, respectively. Finally, we study the synchronization behavior among the neurons using the cross correlation coefficient and the Kuramoto order parameter. We conjecture that unfolding these patterns and behaviors of the network model will help us identify different states of the nervous system, further aiding us in dealing with various neural diseases and nervous disorders.

Global topological synchronization of weighted simplicial complexes

Higher-order networks are able to capture the many-body interactions present in complex systems and to unveil fundamental phenomena revealing the rich interplay between topology, geometry, and dynamics. Simplicial complexes are higher-order networks that encode higher-order topology and dynamics of complex systems. Specifically, simplicial complexes can sustain topological signals, i.e., dynamical variables not only defined on nodes of the network but also on their edges, triangles, and so on. Topological signals can undergo collective phenomena such as synchronization, however, only some higher-order network topologies can sustain global synchronization of topological signals. Here we consider global topological synchronization of topological signals on weighted simplicial complexes. We demonstrate that topological signals can globally synchronize on weighted simplicial complexes, even if they are odd-dimensional, e.g., edge signals, thus overcoming a limitation of the unweighted case. These results thus demonstrate that weighted simplicial complexes are more advantageous for observing these collective phenomena than their unweighted counterpart. In particular, we present two weighted simplicial complexes: the weighted triangulated torus and the weighted waffle. We completely characterize their higher-order spectral properties and demonstrate that, under suitable conditions on their weights, they can sustain global synchronization of edge signals. Our results are interpreted geometrically by showing, among the other results, that in some cases edge weights can be associated with the lengths of the sides of curved simplices.

Reconstructing higher-order interactions in coupled dynamical systems

Higher-order interactions play a key role for the operation and function of a complex system. However, how to identify them is still an open problem. Here, we propose a method to fully reconstruct the structural connectivity of a system of coupled dynamical units, identifying both pairwise and higher-order interactions from the system time evolution. Our method works for any dynamics, and allows the reconstruction of both hypergraphs and simplicial complexes, either undirected or directed, unweighted or weighted. With two concrete applications, we show how the method can help understanding the complexity of bacterial systems, or the microscopic mechanisms of interaction underlying coupled chaotic oscillators.

The transition to synchronization of networked systems

We study the synchronization properties of a generic networked dynamical system, and show that, under a suitable approximation, the transition to synchronization can be predicted with the only help of eigenvalues and eigenvectors of the graph Laplacian matrix. The transition comes out to be made of a well defined sequence of events, each of which corresponds to a specific clustered state. The network's nodes involved in each of the clusters can be identified, and the value of the coupling strength at which the events are taking place can be approximately ascertained. Finally, we present large-scale simulations which show the accuracy of the approximation made, and of our predictions in describing the synchronization transition of both synthetic and real-world large size networks, and we even report that the observed sequence of clusters is preserved in heterogeneous networks made of slightly non-identical systems.

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.

Breathing and switching cyclops states in Kuramoto networks with higher-mode coupling

Cyclops states are intriguing cluster patterns observed in oscillator networks, including neuronal ensembles. The concept of cyclops states formed by two distinct, coherent clusters and a solitary oscillator was introduced by Munyaev et al. [Phys. Rev. Lett. 130, 107201 (2023)0031-900710.1103/PhysRevLett.130.107201], where we explored the surprising prevalence of such states in repulsive Kuramoto networks of rotators with higher-mode harmonics in the coupling. This paper extends our analysis to understand the mechanisms responsible for destroying the cyclops' states and inducing dynamical patterns called breathing and switching cyclops states. We first analytically study the existence and stability of cyclops states in the Kuramoto-Sakaguchi networks of two-dimensional oscillators with inertia as a function of the second coupling harmonic. We then describe two bifurcation scenarios that give birth to breathing and switching cyclops states. We demonstrate that these states and their hybrids are prevalent across a wide coupling range and are robust against a relatively large intrinsic frequency detuning. Beyond the Kuramoto networks, breathing and switching cyclops states promise to strongly manifest in other physical and biological networks, including coupled theta neurons.

Synchronizability in randomized weighted simplicial complexes

We present a formula for determining synchronizability in large, randomized, and weighted simplicial complexes. This formula leverages eigenratios and costs to assess complete synchronizability under diverse network topologies and intensity distributions. We systematically vary coupling strengths (pairwise and three body), degree, and intensity distributions to identify the synchronizability of these simplicial complexes of the identical oscillators with natural coupling. We focus on randomized weighted connections with diffusive couplings and check synchronizability for different cases. For all these scenarios, eigenratios and costs reliably gauge synchronizability, eliminating the need for explicit connectivity matrices and eigenvalue calculations. This efficient approach offers a general formula for manipulating synchronizability in diffusively coupled identical systems with higher-order interactions simply by manipulating degrees, weights, and coupling strengths. We validate our findings with simplicial complexes of Rössler oscillators and confirm that the results are independent of the number of oscillators, connectivity components, and distributions of degrees and intensities. Finally, we validate the theory by considering a real-world connection topology using chaotic Rössler oscillators.

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.

Explosive Cooperation in Social Dilemmas on Higher-Order Networks

Understanding how cooperative behaviors can emerge from competitive interactions is an open problem in biology and social sciences. While interactions are usually modeled as pairwise networks, the units of many real-world systems can also interact in groups of three or more. Here, we introduce a general framework to extend pairwise games to higher-order networks. By studying social dilemmas on hypergraphs with a tunable structure, we find an explosive transition to cooperation triggered by a critical number of higher-order games. The associated bistable regime implies that an initial critical mass of cooperators is also required for the emergence of prosocial behavior. Our results show that higher-order interactions provide a novel explanation for the survival of cooperation.

Strategy evolution on higher-order networks

Cooperation is key to prosperity in human societies. Population structure is well understood as a catalyst for cooperation, where research has focused on pairwise interactions. But cooperative behaviors are not simply dyadic, and they often involve coordinated behavior in larger groups. Here we develop a framework to study the evolution of behavioral strategies in higher-order population structures, which include pairwise and multi-way interactions. We provide an analytical treatment of when cooperation will be favored by higher-order interactions, accounting for arbitrary spatial heterogeneity and nonlinear rewards for cooperation in larger groups. Our results indicate that higher-order interactions can act to promote the evolution of cooperation across a broad range of networks, in public goods games. Higher-order interactions consistently provide an advantage for cooperation when interaction hyper-networks feature multiple conjoined communities. Our analysis provides a systematic account of how higher-order interactions modulate the evolution of prosocial traits.

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.

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.

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.

Adaptation rules inducing synchronization of heterogeneous Kuramoto oscillator network with triadic couplings

A class of adaptation functions is found for which a synchronous mode with different number of phase clusters exists in a network of phase oscillators with triadic couplings. This mode is implemented in a fairly wide range of initial conditions and the maximum number of phase clusters is four. The joint influence of coupling strength and adaptation parameters on synchronization in the network has been studied. The desynchronization transition under variation of the adaptation parameter occurs abruptly and begins with the highest-frequency oscillator, spreading hierarchically to all other elements.

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.

Effect of higher-order interactions on chimera states in two populations of Kuramoto oscillators

We investigate the effect of the fraction of pairwise and higher-order interactions on the emergent dynamics of the two populations of globally coupled Kuramoto oscillators with phase-lag parameters. We find that the stable chimera exists between saddle-node and Hopf bifurcations, while the breathing chimera lives between Hopf and homoclinic bifurcations in the two-parameter phase diagrams. The higher-order interaction facilitates the onset of the bifurcation transitions at a much lower disparity between the inter- and intra-population coupling strengths. Furthermore, the higher-order interaction facilitates the spread of breathing chimera in a large region of the parameter space while suppressing the spread of the stable chimera. A low degree of heterogeneity among the phase-lag parameters promotes the spread of both stable chimera and breathing chimera to a large region of the parameter space for a large fraction of the higher-order coupling. In contrast, a large degree of heterogeneity is found to decrease the spread of both chimera states for a large fraction of the higher-order coupling. A global synchronized state is observed above a critical value of heterogeneity among the phase-lag parameters. We have deduced the low-dimensional evolution equations for the macroscopic order parameters using the Ott-Antonsen Ansatz. We have also deduced the analytical saddle-node and Hopf bifurcation curves from the evolution equations for the macroscopic order parameters and found them to match with the bifurcation curves obtained using the software XPPAUT and with the simulation results.

Higher-order interactions induce anomalous transitions to synchrony

We analyze the simplest model of identical coupled phase oscillators subject to two-body and three-body interactions with permutation symmetry and phase lags. This model is derived from an ensemble of weakly coupled nonlinear oscillators by phase reduction, where the first and second harmonic interactions with phase lags naturally appear. Our study indicates that the higher-order interactions induce anomalous transitions to synchrony. Unlike the conventional Kuramoto model, higher-order interactions lead to anomalous phenomena such as multistability of full synchronization, incoherent, and two-cluster states, and transitions to synchrony through slow switching and clustering. Phase diagrams of the dynamical regimes are constructed theoretically and verified by direct numerical simulations. We also show that similar transition scenarios are observed even if a small heterogeneity in the oscillators' frequency is included.

Emergence and Control of Synchronization in Networks with Directed Many-Body Interactions

The emergence of collective behaviors in networks of dynamical units in pairwise interaction has been explained as the effect of diffusive coupling. How does the presence of higher-order interaction impact the onset of spontaneous or induced synchronous behavior? Inspired by actuation and measurement constraints typical of physical and engineered systems, we propose a diffusion mechanism over hypergraphs that explains the onset of synchronization through a clarifying analogy with signed graphs. Our findings are mathematically backed by general conditions for convergence to the synchronous state.

The complementary contribution of each order topology into the synchronization of multi-order networks

Higher-order interactions improve our capability to model real-world complex systems ranging from physics and neuroscience to economics and social sciences. There is great interest nowadays in understanding the contribution of higher-order terms to the collective behavior of the network. In this work, we investigate the stability of complete synchronization of complex networks with higher-order structures. We demonstrate that the synchronization level of a network composed of nodes interacting simultaneously via multiple orders is maintained regardless of the intensity of coupling strength across different orders. We articulate that lower-order and higher-order topologies work together complementarily to provide the optimal stable configuration, challenging previous conclusions that higher-order interactions promote the stability of synchronization. Furthermore, we find that simply adding higher-order interactions based on existing connections, as in simple complexes, does not have a significant impact on synchronization. The universal applicability of our work lies in the comprehensive analysis of different network topologies, including hypergraphs and simplicial complexes, and the utilization of appropriate rescaling to assess the impact of higher-order interactions on synchronization stability.

Emergence of collective self-oscillations in minimal lattice models with feedback

The emergence of collective oscillations and synchronization is a widespread phenomenon in complex systems. While widely studied in the setting of dynamical systems, this phenomenon is not well understood in the context of out-of-equilibrium phase transitions in many-body systems. Here we consider three classical lattice models, namely the Ising, the Blume-Capel, and the Potts models, provided with a feedback among the order and control parameters. With the help of the linear response theory we derive low-dimensional nonlinear dynamical systems for mean-field cases. These dynamical systems quantitatively reproduce many-body stochastic simulations. In general, we find that the usual equilibrium phase transitions are taken over by more complex bifurcations where nonlinear collective self-oscillations emerge, a behavior that we illustrate by the feedback Landau theory. For the case of the Ising model, we obtain that the bifurcation that takes over the critical point is nontrivial in finite dimensions. Namely, we provide numerical evidence that in two dimensions the most probable value of a cycle's amplitude follows the Onsager law for slow feedback. We illustrate multistability for the case of discontinuously emerging oscillations in the Blume-Capel model, whose tricritical point is substituted by the Bautin bifurcation. For the Potts model with q=3 colors we highlight the appearance of two mirror stable limit cycles at a bifurcation line and characterize the onset of chaotic oscillations that emerge at low temperature through either the Feigenbaum cascade of period doubling or the Afraimovich-Shilnikov scenario of a torus destruction. We also demonstrate that entropy production singularities as a function of the temperature correspond to qualitative change in the spectrum of Lyapunov exponents. Our results show that mean-field collective behavior can be described by the bifurcation theory of low-dimensional dynamical systems, which paves the way for the definition of universality classes of collective oscillations.

First-order transition to oscillation death in coupled oscillators with higher-order interactions

We investigate the dynamical evolution of Stuart-Landau oscillators globally coupled through conjugate or dissimilar variables on simplicial complexes. We report a first-order explosive phase transition from an oscillatory state to oscillation death, with higher-order (2-simplex triadic) interactions, as opposed to the second-order transition with only pairwise (1-simplex) interactions. Moreover, the system displays four distinct homogeneous steady states in the presence of triadic interactions, in contrast to the two homogeneous steady states observed with dyadic interactions. We calculate the backward transition point analytically, confirming the numerical results and providing the origin of the dynamical states in the transition region. The results are robust against the application of noise. The study will be useful in understanding complex systems, such as ecological and epidemiological, having higher-order interactions and coupling through conjugate variables.

Synchronization in simplicial complexes of memristive Rulkov neurons

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

Perfect synchronization in complex networks with higher-order interactions

Achieving perfect synchronization in a complex network, specially in the presence of higher-order interactions (HOIs) at a targeted point in the parameter space, is an interesting, yet challenging task. Here we present a theoretical framework to achieve the same under the paradigm of the Sakaguchi-Kuramoto (SK) model. We analytically derive a frequency set to achieve perfect synchrony at some desired point in a complex network of SK oscillators with higher-order interactions. Considering the SK model with HOIs on top of the scale-free, random, and small world networks, we perform extensive numerical simulations to verify the proposed theory. Numerical simulations show that the analytically derived frequency set not only provides stable perfect synchronization in the network at a desired point but also proves to be very effective in achieving a high level of synchronization around it compared to the other choices of frequency sets. The stability and the robustness of the perfect synchronization state of the system are determined using the low-dimensional reduction of the network and by introducing a Gaussian noise around the derived frequency set, respectively.

Synchronization Induced by Layer Mismatch in Multiplex Networks

Heterogeneity among interacting units plays an important role in numerous biological and man-made complex systems. While the impacts of heterogeneity on synchronization, in terms of structural mismatch of the layers in multiplex networks, has been studied thoroughly, its influence on intralayer synchronization, in terms of parameter mismatch among the layers, has not been adequately investigated. Here, we study the intralayer synchrony in multiplex networks, where the layers are different from one other, due to parameter mismatch in their local dynamics. In such a multiplex network, the intralayer coupling strength for the emergence of intralayer synchronization decreases upon the introduction of impurity among the layers, which is caused by a parameter mismatch in their local dynamics. Furthermore, the area of occurrence of intralayer synchronization also widens with increasing mismatch. We analytically derive a condition under which the intralayer synchronous solution exists, and we even sustain its stability. We also prove that, in spite of the mismatch among the layers, all the layers of the multiplex network synchronize simultaneously. Our results indicate that a multiplex network with mismatched layers can induce synchrony more easily than a multiplex network with identical layers.

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.

Ranking cliques in higher-order complex networks

Traditional network analysis focuses on the representation of complex systems with only pairwise interactions between nodes. However, the higher-order structure, which is beyond pairwise interactions, has a great influence on both network dynamics and function. Ranking cliques could help understand more emergent dynamical phenomena in large-scale complex networks with higher-order structures, regarding important issues, such as behavioral synchronization, dynamical evolution, and epidemic spreading. In this paper, motivated by multi-node interactions in a topological simplex, several higher-order centralities are proposed, namely, higher-order cycle (HOC) ratio, higher-order degree, higher-order H-index, and higher-order PageRank (HOP), to quantify and rank the importance of cliques. Experiments on both synthetic and real-world networks support that, compared with other traditional network metrics, the proposed higher-order centralities effectively reduce the dimension of a large-scale network and are more accurate in finding a set of vital nodes. Moreover, since the critical cliques ranked by the HOP and the HOC are scattered over a complex network, the HOP and the HOC outperform other metrics in ranking cliques that are vital in maintaining the network connectivity, thereby facilitating network dynamical synchronization and virus spread control in applications.

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.

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.