The Kuramoto model in complex networks (1511.07139v1)

Abstract: Synchronization of an ensemble of oscillators is an emergent phenomenon present in several complex systems, ranging from social and physical to biological and technological systems. The most successful approach to describe how coherent behavior emerges in these complex systems is given by the paradigmatic Kuramoto model. This model has been traditionally studied in complete graphs. However, besides being intrinsically dynamical, complex systems present very heterogeneous structure, which can be represented as complex networks. This report is dedicated to review main contributions in the field of synchronization in networks of Kuramoto oscillators. In particular, we provide an overview of the impact of network patterns on the local and global dynamics of coupled phase oscillators. We cover many relevant topics, which encompass a description of the most used analytical approaches and the analysis of several numerical results. Furthermore, we discuss recent developments on variations of the Kuramoto model in networks, including the presence of noise and inertia. The rich potential for applications is discussed for special fields in engineering, neuroscience, physics and Earth science. Finally, we conclude by discussing problems that remain open after the last decade of intensive research on the Kuramoto model and point out some promising directions for future research.

• The paper demonstrates how network topology significantly influences synchronization thresholds using the Kuramoto model.
• It employs analysis of time delays, second-order dynamics, and explosive transitions to elucidate complex synchronization phenomena.
• The study highlights implications for real-world systems like power grids and neural networks by linking structural heterogeneity with phase transitions.

Analysis of the Kuramoto Model in Complex Networks

The Kuramoto model has long been a cornerstone in the paper of synchronization in complex systems. This model, which was originally proposed to understand the synchronous behavior of chemical oscillators, has been extended to diverse fields involving synchronization phenomena, such as neuroscience, power grids, and social dynamics. This essay aims to present an overview of the research on the Kuramoto model when applied to complex networks.

Synchronization and Network Topology

Synchronization within a network of oscillators is significantly influenced by the network's topology. Different types of networks, such as Erdős-Rényi (ER), scale-free (SF), and small-world (SW) networks, show varying dynamics when modeled using Kuramoto oscillators. Researchers have found that highly heterogeneous networks, like SF networks, often exhibit lower synchronization thresholds compared to more homogeneous networks such as ER networks. This is primarily due to the presence of highly connected nodes (hubs) that facilitate synchronization.

Time-delayed Couplings

The introduction of time delays in the interactions between oscillators can yield diverse synchronization outcomes. Time delays are often found in real-world systems where there is a finite time required for information or signals to propagate through the network. The research discussed highlights that time delays can lead to phenomena such as bistability and altered phases of synchronization, where the latter results in different groups of oscillators achieving synchronization at different time delays.

Complex Network Structures

With the growing relevance of realistic network modeling, attention has been given to cluster synchronization, modular networks, and multiplex networks. The presence of clusters or community structure within networks can significantly alter dynamical behavior, leading to possible phase transitions and variations in the critical coupling strength needed for synchronization. In multilayer networks, where nodes participate in multiple types of interactions, the synchronization dynamics can be even more complex, with layer-specific behaviors and interlayer coupling effects needing consideration.

Second-order Kuramoto Model

For certain applications, such as those involving mechanical or electrical systems, the second-order Kuramoto model, which incorporates inertial effects, offers a more accurate depiction of dynamics compared to its first-order counterpart. In these systems, inertia can lead to the slowing down of synchronization and phenomena such as oscillatory transients and chaos.

Explosive Synchronization

Explosive synchronization is a phenomenon where synchronization appears abruptly as the coupling strength crosses a certain threshold. This behavior contrasts with the gradual synchronization seen in traditional Kuramoto models and typically involves a critical correlation between node degrees and natural frequencies within the network. Such behavior has been of interest due to its occurrence in real-world systems such as power grids and neural networks.

Challenges and Future Research

Several challenges persist in the paper of synchronization in complex networks. These include understanding the interplay between network topology and dynamical processes, extending models to account for more realistic features such as dynamic topologies or stochastic effects, and developing efficient numerical and analytical methods to handle the scalability of networks quantitatively.

Future research will likely focus on multi-scale network frameworks, the influence of temporal changes in network structure, and the integration of more complex coupling mechanisms. This will provide deeper insights into not only theoretical aspects but also practical applications in engineering, neuroscience, and social science, thereby advancing the field of complex network dynamics substantially.