Time-delayed Kuramoto model in the Watts-Strogatz small-world networks

Synchronization of two coupled massive oscillators in the time-delayed Kuramoto model

We examine the impact of the time delay on two coupled massive oscillators within the second-order Kuramoto model, which is relevant to the operations of real-world networks that rely on signal transmission speed constraints. Our analytical and numerical exploration shows that time delay can cause multi-stability within phase-locked solutions, and the stability of these solutions decreases as the inertia increases. In addition to phase-locked solutions, we discovered non-phase-locked solutions that exhibit periodic and chaotic behaviors, depending on the amount of inertia and time delay.

Identifying Arnold's tongue for digital oscillators through event-based control in phase-locked loops

Digital phase-locked loops (PLLs) are essential feedback circuits for synchronizing signals in digital communication systems. While amplitude and phase vary continuously in analog oscillators, the amplitude remains constant in digital oscillators with dynamical variations manifesting exclusively through changes in the timing of signal transitions. In this work, we introduce a novel analytically solvable event-based model for phase-locking in digital PLLs that leverages the discrete nature of digital signals. By employing a sampled control strategy, we demonstrate one-to-one and higher ratios of frequency locking under positive and negative feedback. By discretizing the continuous control signal, we drive a discrete iterative map, which we then use to derive analytical expressions for bifurcation curves, analogous to Arnold's tongue in analog oscillators. This mathematical framework provides an analytical approach for the analysis of synchronization and phase-locking in digital oscillators. Furthermore, the event-based control presented in this work for digital oscillators paves the way for energy-efficient circuit design and optimized control strategies for future digital communication systems.

Nonlinear causal network learning via Granger causality based on extreme support vector regression

For complex networked systems, based on the consideration of nonlinearity and causality, a novel general method of nonlinear causal network learning, termed extreme support vector regression Granger causality (ESVRGC), is proposed. The nonuniform time-delayed influence of the driving nodes on the target node is particularly considered. Then, the restricted model and the unrestricted model of Granger causality are, respectively, formulated based on extreme support vector regression, which uses the selected time-delayed components of system variables as the inputs of kernel functions. The nonlinear conditional Granger causality index is finally calculated to confirm the strength of a causal interaction. Generally, based on the simulation of a nonlinear vector autoregressive model and nonlinear discrete time-delayed dynamic systems, ESVRGC demonstrates better performance than other popular methods. Also, the validity and robustness of ESVRGC are also verified by the different cases of network types, sample sizes, noise intensities, and coupling strengths. Finally, the superiority of ESVRGC is successful verified by the experimental study on real benchmark datasets.

Random growth scale-free networked models with an identical degree distribution and a tunable assortativity index

In this work, we propose two kinds of graphic operations by using triangle configuration, based on which we establish a family of random growth networked models G(t;p) where notations t and p represent time step and probability parameter, respectively. By studying some fundamental structural parameters both analytically and numerically, we show that (1) all the realizations G(t;p) follow the same power-law degree distribution with exponent γ=2+ln⁡3/ln⁡2 regardless of probability p and thus have scale-free feature; (2) each model G(t;p) has a relatively high clustering coefficient; and (3) while network G(t;1) has a small average path length, it is not a unique model possessing small-world property mainly because its diameter D(t;1) does not reach the theoretical lower bound. Next, we make use of assortativity index R to quantify the tendency of forming connection between vertices and observe that (1) model G(t;0) exhibits disassortative mixing because the corresponding index R(t;0) is non-positive, and (2) model G(t;1) is in the opposite direction. As a result, we demonstrate that random model G(t;p) has a tunable quantity R(t;p) controlled by probability p. In addition, we exactly determine the total number of spanning trees of deterministic models G(t;1) and G(t;0) and also calculate the entropy of spanning trees.