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Abstract

Most complex systems are nonlinear, relying on emergent behavior resulting from many interacting subsystems, which are often characterized by oscillatory dynamics. Having collective oscillatory behavior is an essential requirement for an appropriate functioning of various real-world systems. Complex networks have proven to be exceptionally efficient in elucidating the topological structures of both natural and artificial systems, as well as describing diverse processes taking place over them. Remarkable advancements have been achieved in recent years in comprehending the emergent dynamics atop complex networks. Specifically, among other processes, a large body of works intend to explore the dynamical robustness of complex networks, which is the networks’ ability to withstand dynamical degradation in the network constituents while maintaining collective oscillatory dynamics. Indeed, various physical and biological systems are recognized to undergo a decline in their dynamic activities, whether occurring naturally or influenced by environmental factors. The impact of such damages on network performance can be significant, and the system’s robustness is indicative of its capability to maintain fundamental functionality in the face of dynamic deteriorations, often called aging. This review offers a comprehensive excerpt of notable research endeavors that scrutinize how networks sustain global oscillation under a growing number of inactive dynamical units. We present the contemporary research dedicated to the theoretical understanding and the enhancement mechanisms of the dynamical robustness in complex networks. Our emphasis lies on various network topologies and coupling functions, elucidating the persistence of networked systems. We cover variants of system characteristics from heterogeneity in network connectivity to heterogeneity in the dynamical units. Finally we discuss challenges ahead in this potential field and open areas for future studies.