Master Stability Functions in Complex Networks

Synchronization is an emergent phenomenon in coupled dynamical networks. The Master Stability Function (MSF) is a highly elegant and powerful tool for characterizing the stability of synchronization states. However, a significant challenge lies in determining the MSF for complex dynamical networks driven by nonlinear interaction mechanisms. These mechanisms introduce additional complexity through the intricate connectivity of interacting elements within the network and the intrinsic dynamics, which are governed by nonlinear processes with diverse parameters and higher dimensionality of systems. Over the past 25 years, extensive research has focused on determining the MSF for pairwise coupled identical systems with diffusive coupling. Our literature survey highlights two significant advancements in recent years: the consideration of multilayer networks instead of single-layer networks and the extension of MSF analysis to incorporate higher-order interactions alongside pairwise interactions. In this review article, we revisit the analysis of the MSF for diffusively pairwise coupled dynamical systems and extend this framework to more general coupling schemes. Furthermore, we systematically derive the MSF for multilayer dynamical networks and single-layer coupled systems by incorporating higher-order interactions alongside pairwise interactions. The primary focus of our review is on the analytical derivation and numerical computation of the MSF for complex dynamical networks. Finally, we demonstrate the application of the MSF in data science, emphasizing its relevance and potential in this rapidly evolving field.