Dynamical robustness of network of oscillators

Most complex systems are nonlinear, relying on emergent behavior from interacting subsystems, often characterized by oscillatory dynamics. Collective oscillatory behavior is essential for the proper functioning of many real world systems. Complex networks have proven efficient in elucidating the topological structures of both natural and artificial systems and describing diverse processes occurring within them. Recent advancements have significantly enhanced our understanding of emergent dynamics in complex networks. Among various processes, a substantial body of work explores the dynamical robustness of complex networks, their ability to withstand degradation in network constituents while maintaining collective oscillatory dynamics. Many physical and biological systems experience a decline in dynamic activities due to natural or environmental factors. The impact of such damages on network performance can be significant, and the system's robustness indicates its capability to maintain functionality despite dynamic changes, often termed aging. This review provides a comprehensive overview of notable research examining how networks sustain global oscillation despite increasing inactive dynamical units. We present contemporary research dedicated to the theoretical understanding and enhancement mechanisms of dynamical robustness in complex networks. Our focus includes various network structures and coupling functions, elucidating the persistence of networked systems. We cover system characteristics from heterogeneity in network connectivity to heterogeneity in dynamical units. Finally, we discuss challenges in this field and open areas for future studies.