  One of the outstanding problems in complexity science and dynamical system
theory is understanding the dynamic behavior of high-dimensional networked
systems and their susceptibility to transitions to undesired states. Because of
varied interactions, large number of parameters and different initial
conditions, the study is extremely difficult and existing methods can be
applied only to continuous-time systems. Here we propose an analytical
framework for collapsing N-dimensional discrete-time systems into a
S+1-dimensional manifold as a function of S effective parameters with S << N.
Specifically, we provide a quantitative prediction of the quality of the
low-dimensional collapse. We test our framework on a variety of real-world
complex systems showing its good performance and correctly identify the regions
in the parameter space corresponding to the system's transitions. Our work
offers an analytical tool to reduce dimensionality of discrete-time networked
systems that can be applied to a broader set of systems and dynamics.
