Chimeric states induced by higher-order interactions in coupled prey-predator systems.

Higher-order interactions have been instrumental in characterizing the intricate complex dynamics in a diverse range of large-scale complex systems. Our study investigates the effect of attractive and repulsive higher-order interactions in globally and non-locally coupled prey-predator Rosenzweig-MacArthur systems. Such interactions lead to the emergence of complex spatiotemporal chimeric states, which are otherwise unobserved in the model system with only pairwise interactions. Our model system exhibits a second-order transition from a chimera-like state (mixture of oscillating and steady state nodes) to a chimera-death state through a supercritical Hopf bifurcation. The origin of these states is discussed in detail along with the effect of the higher-order non-local topology which leads to the rise of a distinct and dynamical state termed as "amplitude-mediated chimera-like states." Our study observes that the introduction of higher-order attractive and repulsive interactions exhibit incoherence and promote persistence in consumer-resource population dynamics as opposed to susceptibility shown by synchronized dynamics with only pairwise interactions, and these results may be of interest to conservationists and theoretical ecologists studying the effect of competing interactions in ecological networks.

Tiered synchronization in Kuramoto oscillators with adaptive higher-order interactions.

Phase transitions widely occur in natural systems. Incorporation of higher-order interactions in coupled dynamics is known to cause first-order phase transition to synchronization in an otherwise smooth second-order in the presence of only pairwise interactions. Here, we discover that adaptation in higher-order interactions restores the second-order phase transition in the former setup and notably produces additional bifurcation referred as tiered synchronization as a consequence of combination of super-critical pitchfork and two saddle node bifurcations. The Ott-Antonsen manifold underlines the interplay of higher-order interactions and adaptation in instigating tiered synchronization, as well as provides complete description of all (un)stable states. These results would be important in comprehending dynamics of real-world systems with inherent higher-order interactions and adaptation through feedback coupling.

Synchronization onset for contrarians with higher-order interactions in multilayer systems.

We investigate the impact of contrarians (via negative coupling) in multilayer networks of phase oscillators having higher-order interactions. We report that the multilayer framework facilitates synchronization onset in the negative pairwise coupling regime. The multilayering strength governs the onset of synchronization and the nature of the phase transition, whereas the higher-order interactions dictate the backward critical coupling. Specifically, the system does not synchronize below a critical value of the multilayering strength. The analytical calculations using the mean-field Ott-Antonsen approach agree with the simulations. The results presented here may be useful for understanding emergent behaviors in real-world complex systems with contrarians and higher-order interactions, such as the brain and social system.

Explosive synchronization in interlayer phase-shifted Kuramoto oscillators on multiplex networks.

Different methods have been proposed in the past few years to incite explosive synchronization (ES) in Kuramoto phase oscillators. In this work, we show that the introduction of a phase shift α in interlayer coupling terms of a two-layer multiplex network of Kuramoto oscillators can also instigate ES in the layers. As α→π/2, ES emerges along with hysteresis. The width of hysteresis depends on the phase shift α, interlayer coupling strength, and natural frequency mismatch between mirror nodes. A mean-field analysis is performed to justify the numerical results. Similar to earlier works, the suppression of synchronization is accountable for the occurrence of ES. The robustness of ES against changes in network topology and natural frequency distribution is tested. Finally, taking a suggestion from the synchronized state of the multiplex networks, we extend the results to classical single networks where some specific links are assigned phase-shifted interactions.

Synchronization in multiplex models of neuron-glial systems: Small-world topology and inhibitory coupling.

In this work, we investigate the impact of mixed coupling on synchronization in a multiplex oscillatory network. The network mimics the neural-glial systems by incorporating interacting slow ("glial") and fast ("neural") oscillatory layers. Connections between the "glial" elements form a regular periodic structure, in which each element is connected to the eight other neighbor elements, whereas connections among "neural" elements are represented by Watts-Strogatz networks (from regular and small-world to random Erdös-Rényi graph) with a matching mean node degree. We find that the random rewiring toward small-world topology readily yields the dynamics close to that exhibited for a completely random graph, in particular, leading to coarse-graining of dynamics, suppressing multi-stability of synchronization regimes, and the onset of Kuramoto-type synchrony in both layers. The duration of transient dynamics in the system measured by relaxation times is minimized with the increase of random connections in the neural layer, remaining substantial only close to synchronization-desynchronization transitions. "Inhibitory" interactions in the "neural" subnetwork layer undermine synchronization; however, the strong coupling with the "glial" layer overcomes this effect.

Catalytic feed-forward explosive synchronization in multilayer networks.

Inhibitory couplings are crucial for the normal functioning of many real-world complex systems. Inhibition in one layer has been shown to induce explosive synchronization in another excitatory (or positive) layer of duplex networks. By extending this framework to multiplex networks, this article shows that inhibition in a single layer can act as a catalyst, leading to explosive synchronization transitions in the rest of the layers feed-forwarded through intermediate layer(s). Considering a multiplex network of coupled Kuramoto oscillators, we demonstrate that the characteristics of the transition emergent in a layer can be entirely controlled by the intra-layer coupling of other layers and the multiplexing strengths. The results presented here are essential to fathom the synchronization behavior of coupled dynamical units in multi-layer systems possessing inhibitory coupling in one of its layers, representing the importance of multiplexing.

Machine learning assisted network classification from symbolic time-series.

Machine learning techniques have been witnessing perpetual success in predicting and understanding behaviors of a diverse range of complex systems. By employing a deep learning method on limited time-series information of a handful of nodes from large-size complex systems, we label the underlying network structures assigned in different classes. We consider two popular models, namely, coupled Kuramoto oscillators and susceptible-infectious-susceptible to demonstrate our results. Importantly, we elucidate that even binary information of the time evolution behavior of a few coupled units (nodes) yields as accurate classification of the underlying network structure as achieved by the actual time-series data. The key of the entire process reckons on feeding the time-series information of the nodes when the system evolves in a partially synchronized state, i.e., neither completely incoherent nor completely synchronized. The two biggest advantages of our method over previous existing methods are its simplicity and the requirement of the time evolution of one largest degree node or a handful of the nodes to predict the classification of large-size networks with remarkable accuracy.

Identification of chimera using machine learning.

Chimera state refers to the coexistence of coherent and non-coherent phases in identically coupled dynamical units found in various complex dynamical systems. Identification of chimera, on one hand, is essential due to its applicability in various areas including neuroscience and, on the other hand, is challenging due to its widely varied appearance in different systems and the peculiar nature of its profile. Therefore, a simple yet universal method for its identification remains an open problem. Here, we present a very distinctive approach using machine learning techniques to characterize different dynamical phases and identify the chimera state from given spatial profiles generated using various different models. The experimental results show that the performance of the classification algorithms varies for different dynamical models. The machine learning algorithms, namely, random forest, oblique random forest based on Tikhonov, axis-parallel split, and null space regularization achieved more than 96% accuracy for the Kuramoto model. For the logistic maps, random forest and Tikhonov regularization based oblique random forest showed more than 90% accuracy, and for the Hénon map model, random forest, null space, and axis-parallel split regularization based oblique random forest achieved more than 80% accuracy. The oblique random forest with null space regularization achieved consistent performance (more than 83% accuracy) across different dynamical models while the auto-encoder based random vector functional link neural network showed relatively lower performance. This work provides a direction for employing machine learning techniques to identify dynamical patterns arising in coupled non-linear units on large-scale and for characterizing complex spatiotemporal patterns in real-world systems for various applications.

Explosive synchronization in frequency displaced multiplex networks.

Motivated by the recent multiplex framework of complex networks, in this work, we investigate if explosive synchronization can be induced in the multiplex network of two layers. Using nonidentical Kuramoto oscillators, we show that a sufficient frequency mismatch between two layers of a multiplex network can lead to explosive inter- and intralayer synchronization due to mutual frustration in the completion of the synchronization processes of the layers, generating a hybrid transition without imposing any specific structure-dynamics correlation.

Weak multiplexing in neural networks: Switching between chimera and solitary states.

We investigate spatio-temporal patterns occurring in a two-layer multiplex network of oscillatory FitzHugh-Nagumo neurons, where each layer is represented by a nonlocally coupled ring. We show that weak multiplexing, i.e., when the coupling between the layers is smaller than that within the layers, can have a significant impact on the dynamics of the neural network. We develop control strategies based on weak multiplexing and demonstrate how the desired state in one layer can be achieved without manipulating its parameters, but only by adjusting the other layer. We find that for coupling range mismatch, weak multiplexing leads to the appearance of chimera states with different shapes of the mean velocity profile for parameter ranges where they do not exist in isolation. Moreover, we show that introducing a coupling strength mismatch between the layers can suppress chimera states with one incoherent domain (one-headed chimeras) and induce various other regimes such as in-phase synchronization or two-headed chimeras. Interestingly, small intra-layer coupling strength mismatch allows to achieve solitary states throughout the whole network.

Spectral properties of complex networks.

This review presents an account of the major works done on spectra of adjacency matrices drawn on networks and the basic understanding attained so far. We have divided the review under three sections: (a) extremal eigenvalues, (b) bulk part of the spectrum, and (c) degenerate eigenvalues, based on the intrinsic properties of eigenvalues and the phenomena they capture. We have reviewed the works done for spectra of various popular model networks, such as the Erdos-Rényi random networks, scale-free networks, 1-d lattice, small-world networks, and various different real-world networks. Additionally, potential applications of spectral properties for natural processes have been reviewed.

Engineering chimera patterns in networks using heterogeneous delays.

Symmetry breaking spatial patterns, referred to as chimera states, have recently been catapulted into the limelight due to their coexisting coherent and incoherent hybrid dynamics. Here, we present a method to engineer a chimera state by using an appropriate distribution of heterogeneous time delays on the edges of a network. The time delays in interactions, intrinsic to natural or artificial complex systems, are known to induce various modifications in spatiotemporal behaviors of the coupled dynamics on networks. Using a coupled chaotic map with the identical coupling environment, we demonstrate that control over the spatial location of the incoherent region of a chimera state in a network can be achieved by appropriately introducing time delays. This method allows for the engineering of tailor-made one cluster or multi-cluster chimera patterns. Furthermore, borrowing a measure of eigenvector localization from the spectral graph theory, we introduce a spatial inverse participation ratio, which provides a robust way for the identification of the chimera state. This report highlights the necessity to consider the heterogeneous time delays to develop applications for the chimera states in particular and understand coupled dynamical systems in general.

Is repulsion good for the health of chimeras?

Yes! Very much so. A chimera state refers to the coexistence of a coherent-incoherent dynamical evolution of identically coupled oscillators. We investigate the impact of multiplexing of a layer having repulsively coupled oscillators on the occurrence of chimeras in the layer having attractively coupled identical oscillators. We report that there exists an enhancement in the appearance of the chimera state in one layer of the multiplex network in the presence of repulsive coupling in the other layer. Furthermore, we show that a small amount of inhibition or repulsive coupling in one layer is sufficient to yield the chimera state in another layer by destroying its synchronized behavior. These results can be used to obtain insight into dynamical behaviors of those systems where both attractive and repulsive couplings exist among their constituents.

Interplay of delay and multiplexing: Impact on cluster synchronization.

Communication delays and multiplexing are ubiquitous features of real-world network systems. We here introduce a simple model where these two features are simultaneously present and report the rich phenomenology which is actually due to their interplay on cluster synchronization. A delay in one layer has non trivial impacts on the collective dynamics of the other layers, enhancing or suppressing synchronization. At the same time, multiplexing may also enhance cluster synchronization of delayed layers. We elucidate several nontrivial (and anti-intuitive) scenarios, which are of interest and potential application in various real-world systems, where the introduction of a delay may render synchronization of a layer robust against changes in the properties of the other layers.

Origin and implications of zero degeneracy in networks spectra.

The spectra of many real world networks exhibit properties which are different from those of random networks generated using various models. One such property is the existence of a very high degeneracy at the zero eigenvalue. In this work, we provide all the possible reasons behind the occurrence of the zero degeneracy in the network spectra, namely, the complete and partial duplications, as well as their implications. The power-law degree sequence and the preferential attachment are the properties which enhances the occurrence of such duplications and hence leading to the zero degeneracy. A comparison of the zero degeneracy in protein-protein interaction networks of six different species and in their corresponding model networks indicates importance of the degree sequences and the power-law exponent for the occurrence of zero degeneracy.

Prolonged hysteresis in the Kuramoto model with inertia and higher-order interactions.

The inclusion of inertia in the Kuramoto model has long been reported to change the nature of a phase transition, providing a fertile ground to model the dynamical behaviors of interacting units. More recently, higher-order interactions have been realized as essential for the functioning of real-world complex systems ranging from the brain to disease spreading. Yet analytical insights to decipher the role of inertia with higher-order interactions remain challenging. Here, we study the Kuramoto model with inertia on simplicial complexes, merging two research domains. We develop an analytical framework in a mean-field setting using self-consistent equations to describe the steady-state behavior, which reveals a prolonged hysteresis in the synchronization profile. Inertia and triadic interaction strength exhibit isolated influence on system dynamics by predominantly governing, respectively, the forward and backward transition points. This paper sets a paradigm to deepen our understanding of real-world complex systems such as power grids modeled as the Kuramoto model with inertia.

Rotating clusters in phase-lagged Kuramoto oscillators with higher-order interactions.

The effect of phase-lag in pairwise interactions has been a topic of great interest for a while. However, real-world systems often have interactions that are beyond pairwise and can be modeled using simplicial complexes. We show that the inclusion of higher-order interactions in phase-lagged coupled Kuramoto oscillators shifts the critical point at which first-order transition from a cluster synchronized state to an incoherent state takes place. Considering the polar coordinates, we obtain the rotation frequency of the clusters, which turns out to be a function of the phase-lag parameter. In turn, the phase- lag can be used as a control parameter to achieve a desired cluster frequency. Moreover, in the thermodynamic limit, by employing the Ott-Antonsen approach we derive a reduced equation for the order parameter measuring cluster synchronization and progress further through the self-consistency method to obtain a closed form of the order parameter measuring global synchronization which was lacking in the Ott-Antonsen approach.

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.

Solitary death in coupled limit cycle oscillators with higher-order interactions.

Coupled limit cycle oscillators with pairwise interactions are known to depict phase transitions from an oscillatory state to amplitude or oscillation death. This Research Letter introduces a scheme to incorporate higher-order interactions which cannot be decomposed into pairwise interactions and investigates the dynamical evolution of Stuart-Landau oscillators under the impression of such a coupling. We discover an oscillator death state through a first-order (explosive) phase transition in which a single, coupling-dependent stable death state away from the origin exists in isolation without being accompanied by any other stable state usually existing for pairwise couplings. We call such a state a solitary death state. Contrary to widespread subcritical Hopf bifurcation, here we report homoclinic bifurcation as an origin of the explosive death state. Moreover, this explosive transition to the death state is preceded by a surge in amplitude and followed by a revival of the oscillations. The analytical value of the critical coupling strength for the solitary death state agrees with the simulation results. Finally, we point out the resemblance of the results with different dynamical states associated with epileptic seizures.

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.