Emergent explosive synchronization in adaptive complex networks

Synchronization transitions in phase oscillator populations with partial adaptive coupling

The adaptation underlying many realistic processes plays a pivotal role in shaping the collective dynamics of diverse systems. Here, we untangle the generic conditions for synchronization transitions in a system of coupled phase oscillators incorporating the adaptive scheme encoded by the feedback between the coupling and the order parameter via a power-law function with different weights. We mathematically argue that, in the subcritical and supercritical correlation scenarios, there exists no critical adaptive fraction for synchronization transitions converting from the first (second)-order to the second (first)-order. In contrast to the synchronization transitions previously deemed, the explosive and continuous phase transitions take place in the corresponding regions as long as the adaptive fraction is nonzero, respectively. Nevertheless, we uncover that, at the critical correlation, the routes toward synchronization depend crucially on the relative adaptive weights. In particular, we unveil that the emergence of a range of interrelated scaling behaviors of the order parameter near criticality, manifesting the subcritical and supercritical bifurcations, are responsible for various observed phase transitions. Our work, thus, provides profound insights for understanding the dynamical nature of phase transitions, and for better controlling and manipulating synchronization transitions in networked systems with adaptation.

Effect of adaptation functions and multilayer topology on synchronization

This study investigates the synchronization of globally coupled Kuramoto oscillators in monolayer and multilayer configurations. The interactions are taken to be pairwise, whose strength adapts with the instantaneous synchronization order parameter. The route to synchronization is analytically investigated using the Ott-Antonsen ansatz for two broad classes of adaptation functions that capture a wide range of transition scenarios. The formulation is subsequently extended to adaptively coupled multilayer configurations, using which a wider range of transition scenarios is uncovered for a bilayer model with cross-adaptive interlayer interactions.

Hebbian and anti-Hebbian adaptation-induced dynamical states in adaptive networks

We investigate the interplay of an external forcing and an adaptive network, whose connection weights coevolve with the dynamical states of the phase oscillators. In particular, we consider the Hebbian and anti-Hebbian adaptation mechanisms for the evolution of the connection weights. The Hebbian adaptation manifests several interesting partially synchronized states, such as phase and frequency clusters, bump state, bump frequency phase clusters, and forced entrained clusters, in addition to the completely synchronized and forced entrained states. Anti-Hebbian adaptation facilitates the manifestation of the itinerant chimera characterized by randomly evolving coherent and incoherent domains along with some of the aforementioned dynamical states induced by the Hebbian adaptation. We introduce three distinct measures for the strength of incoherence based on the local standard deviations of the time-averaged frequency and the instantaneous phase of each oscillator, and the time-averaged mean frequency for each bin to corroborate the distinct dynamical states and to demarcate the two parameter phase diagrams. We also arrive at the existence and stability conditions for the forced entrained state using the linear stability analysis, which is found to be consistent with the simulation results.

Perspectives on adaptive dynamical systems

Adaptivity is a dynamical feature that is omnipresent in nature, socio-economics, and technology. For example, adaptive couplings appear in various real-world systems, such as the power grid, social, and neural networks, and they form the backbone of closed-loop control strategies and machine learning algorithms. In this article, we provide an interdisciplinary perspective on adaptive systems. We reflect on the notion and terminology of adaptivity in different disciplines and discuss which role adaptivity plays for various fields. We highlight common open challenges and give perspectives on future research directions, looking to inspire interdisciplinary approaches.

Heterogeneous Nucleation in Finite-Size Adaptive Dynamical Networks

Phase transitions in equilibrium and nonequilibrium systems play a major role in the natural sciences. In dynamical networks, phase transitions organize qualitative changes in the collective behavior of coupled dynamical units. Adaptive dynamical networks feature a connectivity structure that changes over time, coevolving with the nodes' dynamical state. In this Letter, we show the emergence of two distinct first-order nonequilibrium phase transitions in a finite-size adaptive network of heterogeneous phase oscillators. Depending on the nature of defects in the internal frequency distribution, we observe either an abrupt single-step transition to full synchronization or a more gradual multistep transition. This observation has a striking resemblance to heterogeneous nucleation. We develop a mean-field approach to study the interplay between adaptivity and nodal heterogeneity and describe the dynamics of multicluster states and their role in determining the character of the phase transition. Our work provides a theoretical framework for studying the interplay between adaptivity and nodal heterogeneity.

First-order route to antiphase clustering in adaptive simplicial complexes

This Letter investigates the transition to synchronization of oscillator ensembles encoded by simplicial complexes in which pairwise and higher-order coupling weights alter with time through a rate-based adaptive mechanism inspired by the Hebbian learning rule. These simultaneously evolving disparate adaptive coupling weights lead to a phenomenon in that the in-phase synchronization is completely obliterated; instead, the antiphase synchronization is originated. In addition, the onsets of antiphase synchronization and desynchronization are manageable through both dyadic and triadic learning rates. The theoretical validation of these numerical assessments is delineated thoroughly by employing Ott-Antonsen dimensionality reduction. The framework and results of the Letter would help understand the underlying synchronization behavior of a range of real-world systems, such as the brain functions and social systems where interactions evolve with time.

Exotic states induced by coevolving connection weights and phases in complex networks

We consider an adaptive network, whose connection weights coevolve in congruence with the dynamical states of the local nodes that are under the influence of an external stimulus. The adaptive dynamical system mimics the adaptive synaptic connections common in neuronal networks. The adaptive network under external forcing displays exotic dynamical states such as itinerant chimeras whose population density of coherent and incoherent domains coevolves with the synaptic connection, bump states, and bump frequency cluster states, which do not exist in adaptive networks without forcing. In addition, the adaptive network also exhibits partial synchronization patterns such as phase and frequency clusters, forced entrained, and incoherent states. We introduce two measures for the strength of incoherence based on the standard deviation of the temporally averaged (mean) frequency and on the mean frequency to classify the emergent dynamical states as well as their transitions. We provide a two-parameter phase diagram showing the wealth of dynamical states. We additionally deduce the stability condition for the frequency-entrained state. We use the paradigmatic Kuramoto model of phase oscillators, which is a simple generic model that has been widely employed in unraveling a plethora of cooperative phenomena in natural and man-made systems.

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.

Explosive synchronization and chimera in interpinned multilayer networks

This Letter investigates the nature of synchronization in multilayered and multiplexed populations in which the interlayer interactions are randomly pinned. First, we show that a multilayer network constructed by setting up all-to-all interlayer connections between the two populations leads to explosive synchronization in the two populations successively, leading to the coexistence of coherent and incoherent populations forming chimera states. Second, a multiplex formation of the two populations in which only the mirror nodes are interconnected espouses explosive transitions in the two populations concurrently. The occurrence of both explosive synchronization and chimera are substantiated with rigorous theoretical mean-field analysis. The random pinning in the interlayer interactions concerns the practical problems where the impact of dynamics of one network on that of other interconnected networks remains elusive, as is the case for many real-world systems.

Assortativity-induced explosive synchronization in a complex neuronal network

In this study, we consider a scale-free network of nonidentical Chialvo neurons, coupled through electrical synapses. For sufficiently strong coupling, the system undergoes a transition from completely out of phase synchronized to phase synchronized state. The principal focus of this study is to investigate the effect of the degree of assortativity over the synchronization transition process. It is observed that, depending on assortativity, bistability between two asymptotically stable states allows one to develop a hysteresis loop which transforms the phase transition from second order to first order. An expansion in the area of hysteresis loop is noticeable with increasing degree-degree correlation in the network. Our study also reveals that effective frequencies of nodes simultaneously go through a continuous or sudden transition to the synchronized state with the corresponding phases. Further, we examine the robustness of the results under the effect of network size and average degree, as well as diverse frequency setup. Finally, we investigate the dynamical mechanism in the process of generating explosive synchronization. We observe a significant impact of lower degree nodes behind such phenomena: in a positive assortative network the low degree nodes delay the synchronization transition.

Transition to synchronization in heterogeneous inhibitory neural networks with structured synapses

Inhibitory neurons form an extensive network involved in the development of different rhythms in the cerebral cortex. A transition from an incoherent state, where all inhibitory neurons fire unrelated to each other, to a synchronized or locked state, where all or most neurons define a tight firing pattern, is maybe the most salient process to analyze when considering neuronal rhythms. In this work, we analyzed whether different patterns of effective synaptic connectivity may support a first-order-like transition in this path to synchronization. Such an "explosive" phenomenon may be relevant in neural processes, as normal cognitive processing in different tasks and some neurological disorders manifest an increased power in many neuronal rhythms, supported by an extended concerted spiking activity and an abrupt change to this state. Furthermore, we built an adaptive mechanism that supports the generation of this kind of network, which rapidly creates the underlying structure based on the ongoing firing statistics.

Stability of synchronization in simplicial complexes

Various systems in physics, biology, social sciences and engineering have been successfully modeled as networks of coupled dynamical systems, where the links describe pairwise interactions. This is, however, too strong a limitation, as recent studies have revealed that higher-order many-body interactions are present in social groups, ecosystems and in the human brain, and they actually affect the emergent dynamics of all these systems. Here, we introduce a general framework to study coupled dynamical systems accounting for the precise microscopic structure of their interactions at any possible order. We show that complete synchronization exists as an invariant solution, and give the necessary condition for it to be observed as a stable state. Moreover, in some relevant instances, such a necessary condition takes the form of a Master Stability Function. This generalizes the existing results valid for pairwise interactions to the case of complex systems with the most general possible architecture.

Co-adaptation enhances the resilience of mutualistic networks

Mutualistic networks, which describe the ecological interactions between multiple types of species such as plants and pollinators, play a paramount role in the generation of Earth's biodiversity. The resilience of a mutualistic network denotes its ability to retain basic functionality when errors and failures threaten the persistence of the community. Under the disturbances of mass extinctions and human-induced disasters, it is crucial to understand how mutualistic networks respond to changes, which enables the system to increase resilience and tolerate further damages. Despite recent advances in the modelling of the structure-based adaptation, we lack mathematical and computational models to describe and capture the co-adaptation between the structure and dynamics of mutualistic networks. In this paper, we incorporate dynamic features into the adaptation of structure and propose a co-adaptation model that drastically enhances the resilience of non-adaptive and structure-based adaptation models. Surprisingly, the reason for the enhancement is that the co-adaptation mechanism simultaneously increases the heterogeneity of the mutualistic network significantly without changing its connectance. Owing to the broad applications of mutualistic networks, our findings offer new ways to design mechanisms that enhance the resilience of many other systems, such as smart infrastructures and social-economical systems.

Birth and Stabilization of Phase Clusters by Multiplexing of Adaptive Networks

We propose a concept to generate and stabilize diverse partial synchronization patterns (phase clusters) in adaptive networks which are widespread in neuroscience and social sciences, as well as biology, engineering, and other disciplines. We show by theoretical analysis and computer simulations that multiplexing in a multilayer network with symmetry can induce various stable phase cluster states in a situation where they are not stable or do not even exist in the single layer. Further, we develop a method for the analysis of Laplacian matrices of multiplex networks which allows for insight into the spectral structure of these networks enabling a reduction to the stability problem of single layers. We employ the multiplex decomposition to provide analytic results for the stability of the multilayer patterns. As local dynamics we use the paradigmatic Kuramoto phase oscillator, which is a simple generic model and has been successfully applied in the modeling of synchronization phenomena in a wide range of natural and technological systems.

Hierarchical frequency clusters in adaptive networks of phase oscillators

Adaptive dynamical networks appear in various real-word systems. One of the simplest phenomenological models for investigating basic properties of adaptive networks is the system of coupled phase oscillators with adaptive couplings. In this paper, we investigate the dynamics of this system. We extend recent results on the appearance of hierarchical frequency multiclusters by investigating the effect of the time scale separation. We show that the slow adaptation in comparison with the fast phase dynamics is necessary for the emergence of the multiclusters and their stability. Additionally, we study the role of double antipodal clusters, which appear to be unstable for all considered parameter values. We show that such states can be observed for a relatively long time, i.e., they are metastable. A geometrical explanation for such an effect is based on the emergence of a heteroclinic orbit.

Inhibition-induced explosive synchronization in multiplex networks

To date, explosive synchronization (ES) in a network is shown to be originated from considering either degree-frequency correlation, frequency-coupling strength correlation, inertia, or adaptively controlled phase oscillators. Here we show that ES is a generic phenomenon and can occur in any network by multiplexing it with an appropriate layer without even considering any prerequisite for the emergence of ES. We devise a technique which leads to the occurrence of ES with hysteresis loop in a network upon its multiplexing with a negatively coupled (or inhibitory) layer. The impact of various structural properties of positively coupled (or excitatory) and inhibitory layers along with the strength of multiplexing in gaining control over the induced ES transition is discussed. Analytical prediction for the spread of phase distribution of each layer is provided, which is in good agreement with the numerical assessment. This investigation is a step forward in highlighting the importance of multiplex framework not only in bringing phenomena which are not possible in an isolated network but also in providing more structural control over the induced phenomena.