  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.
