Transition to synchrony in degree-frequency correlated Sakaguchi-Kuramoto model

Impact of phase lag on synchronization in frustrated Kuramoto model with higher-order interactions

The study of first order transition (explosive synchronization) in an ensemble (network) of coupled oscillators has been the topic of paramount interest among the researchers for more than one decade. Several frameworks have been proposed to induce explosive synchronization in a network and it has been reported that phase frustration in a network usually suppresses first order transition in the presence of pairwise interactions among the oscillators. However, on the contrary, by considering networks of phase frustrated coupled oscillators in the presence of higher-order interactions (up to 2-simplexes) we show here, under certain conditions, phase frustration can promote explosive synchronization in a network. A low-dimensional model of the network in the thermodynamic limit is derived using the Ott-Antonsen ansatz to explain this surprising result. Analytical treatment of the low-dimensional model, including bifurcation analysis, explains the apparent counter intuitive result quite clearly.

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.

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.

Extreme events in a complex network: Interplay between degree distribution and repulsive interaction

The role of topological heterogeneity in the origin of extreme events in a network is investigated here. The dynamics of the oscillators associated with the nodes are assumed to be identical and influenced by mean-field repulsive interactions. An interplay of topological heterogeneity and the repulsive interaction between the dynamical units of the network triggers extreme events in the nodes when each node succumbs to such events for discretely different ranges of repulsive coupling. A high degree node is vulnerable to weaker repulsive interactions, while a low degree node is susceptible to stronger interactions. As a result, the formation of extreme events changes position with increasing strength of repulsive interaction from high to low degree nodes. Extreme events at any node are identified with the appearance of occasional large-amplitude events (amplitude of the temporal dynamics) that are larger than a threshold height and rare in occurrence, which we confirm by estimating the probability distribution of all events. Extreme events appear at any oscillator near the boundary of transition from rotation to libration at a critical value of the repulsive coupling strength. To explore the phenomenon, a paradigmatic second-order phase model is used to represent the dynamics of the oscillator associated with each node. We make an annealed network approximation to reduce our original model and, thereby, confirm the dual role of the repulsive interaction and the degree of a node in the origin of extreme events in any oscillator associated with a node.

Mean-field theory for double-well systems on degree-heterogeneous networks

Many complex dynamical systems in the real world, including ecological, climate, financial and power-grid systems, often show critical transitions, or tipping points, in which the system’s dynamics suddenly transit into a qualitatively different state. In mathematical models, tipping points happen as a control parameter gradually changes and crosses a certain threshold. Tipping elements in such systems may interact with each other as a network, and understanding the behaviour of interacting tipping elements is a challenge because of the high dimensionality originating from the network. Here, we develop a degree-based mean-field theory for a prototypical double-well system coupled on a network with the aim of understanding coupled tipping dynamics with a low-dimensional description. The method approximates both the onset of the tipping point and the position of equilibria with a reasonable accuracy. Based on the developed theory and numerical simulations, we also provide evidence for multistage tipping point transitions in networks of double-well 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.

Amplification of explosive width in complex networks

We present an adaptive coupling strategy to induce hysteresis/explosive synchronization in complex networks of phase oscillators (Sakaguchi-Kuramoto model). The coupling strategy ensures explosive synchronization with significant explosive width enhancement. Results show the robustness of the strategy, and the strategy can diminish (by inducing enhanced hysteresis loop) the contrarian impact of phase frustration in the network, irrespective of the network structure or frequency distributions. Additionally, we design a set of frequency for the oscillators, which eventually ensure complete in-phase synchronization behavior among these oscillators (with enhanced explosive width) in the case of adaptive-coupling scheme. Based on a mean-field analysis, we develop a semi-analytical formalism, which can accurately predict the backward transition of the synchronization order parameter.

Synchronization in starlike networks of phase oscillators

We fully describe the mechanisms underlying synchronization in starlike networks of phase oscillators. In particular, the routes to synchronization and the critical points for the associated phase transitions are determined analytically. In contrast to the classical Kuramoto theory, we unveil that relaxation rates to each equilibrium state indeed exist and remain invariant under three levels of descriptions corresponding to different geometric implications. The special symmetry in the coupling determines a quasi-Hamiltonian property, which is further unveiled on the basis of singular perturbation theory. Since starlike coupling configurations constitute the building blocks of technological and biological real world networks, our paper paves the way towards the understanding of the functioning of such real world systems in many practical situations.

Repulsive synchronization in complex networks

Synchronization in complex networks characterizes what happens when an ensemble of oscillators in a complex autonomous system become phase-locked. We study the Kuramoto model with a tunable phase-lag parameter α in the coupling term to determine how phase shifts influence the synchronization transition. The simulation results show that the phase frustration parameter leads to desynchronization. We find two global synchronization regions for α∈[0,2π) when the coupling is sufficiently large and detect a relatively rare network synchronization pattern in the frustration parameter near α=π. We call this frequency-locking configuration as "repulsive synchronization," because it is induced by repulsive coupling. Since the repulsive synchronization cannot be described by the usual order parameter r, the parameter frequency dispersion is introduced to detect synchronization.

Synchronization transition in Sakaguchi-Kuramoto model on complex networks with partial degree-frequency correlation

We investigate transition to synchronization in the Sakaguchi-Kuramoto (SK) model on complex networks analytically as well as numerically. Natural frequencies of a percentage (f) of higher degree nodes of the network are assumed to be correlated with their degrees and that of the remaining nodes are drawn from some standard distribution, namely, Lorentz distribution. The effects of variation of f and phase frustration parameter α on transition to synchronization are investigated in detail. Self-consistent equations involving critical coupling strength (λ_c) and group angular velocity (Ω_c) at the onset of synchronization have been derived analytically in the thermodynamic limit. For the detailed investigation, we considered the SK model on scale-free (SF) as well as Erdős-Rényi (ER) networks. Interestingly, explosive synchronization (ES) has been observed in both networks for different ranges of values of α and f. For SF networks, as the value of f is set within 10%≤f≤70%, the range of the values of α for existence of the ES is greatly enhanced compared to the fully degree-frequency correlated case when scaling exponent γ<3. ES is also observed in SF networks with γ>3, which is never observed in fully degree-frequency correlated environment. On the other hand, for random networks, ES observed is in a narrow window of α when the value of f is taken within 30%≤f≤50%. In all the cases, critical coupling strengths for transition to synchronization computed from the analytically derived self-consistent equations show a very good agreement with the numerical results. Finally, we observe ES in the metabolic network of the roundworm Caenorhabditis elegans in partially degree-frequency correlated environment.