Coarse-graining of Dynamics in Weakly Non-linear Processes on Networks

It is known that if the graph has a hierarchical structure, many dynamical processes exhibit transient states in which the dynamics synchronizes or stabilizes at communities in one level of hierarchy.
In linear and weakly non-linear processes these transient states appear at time scales associated with inverse of the so-called community eigenvectors of the quadratic piece of the Hamiltonian.
We show that these transient states can be exploited to coarse-grain the network during the dynamics and reduce the time complexity of estimating the outcome of a dynamical process from $O(N^3)$ down to a lower complexity in most hierarchical networks, and possibly to $O(N^2\log N)$ in tree-like networks.
We outline the coarse-graining procedure, which relies on spectral methods and can be computed efficiently.
We also discuss potential relations with graph learning and regression on graphs.