  Dimension reduction is a common strategy to study non-linear dynamical
systems composed by a large number of variables. The goal is to find a smaller
version of the system whose time evolution is easier to predict while
preserving some of the key dynamical features of the original system. Finding
such a reduced representation for complex systems is, however, a difficult
task. We address this problem for dynamics on weighted directed networks, with
special emphasis on modular and heterogeneous networks. We propose a two-step
dimension-reduction method that takes into account the properties of the
adjacency matrix. First, units are partitioned into groups of similar
connectivity profiles. Each group is associated to an observable that is a
weighted average of the nodes' activities within the group. Second, we derive a
set of conditions that must be fulfilled for these observables to properly
represent the original system's behavior, together with a method for
approximately solving them. The result is a reduced adjacency matrix and an
approximate system of ODEs for the observables' evolution. We show that the
reduced system can be used to predict some characteristic features of the
complete dynamics for different types of connectivity structures, both
synthetic and derived from real data, including neuronal, ecological, and
social networks. Our formalism opens a way to a systematic comparison of the
effect of various structural properties on the overall network dynamics. It can
thus help to identify the main structural driving forces guiding the evolution
of dynamical processes on networks.
