  Networks of interconnected agents are essential to study complex networked
systems' state evolution, stability, resilience, and control. Nevertheless, the
high dimensionality and nonlinear dynamics are vital factors preventing us from
theoretically analyzing them. Recently, the dimension-reduction approaches
reduced the system's size by mapping the original system to a one-dimensional
system such that only one effective representative can capture its macroscopic
dynamics. However, the approaches dramatically fail as the network becomes
heterogeneous and has multiple community structures. Here, we bridge the gap by
developing a generalized dimension reduction approach, which enables us to map
the original system to a $m$-dimensional system that consists of $m$
interacting components. Notably, by validating it on various dynamical models,
this approach accurately predicts the original system state and the tipping
point, if any. Furthermore, the numerical results demonstrate that this
approach approximates the system evolution and identifies the critical points
for complex networks with time delay.
