  Graph theory constitutes a widely used and established field providing
powerful tools for the characterization of complex networks. The intricate
topology of networks can also be investigated by means of the collective
dynamics observed in the interactions of self-sustained oscillations
(synchronization patterns) or propagation-like processes such as random walks.
However, networks are often inferred from real data forming dynamic systems,
which are different from those employed to reveal their topological
characteristics. This stresses the necessity for a theoretical framework
dedicated to the mutual relationship between the structure and dynamics in
complex networks, as the two sides of the same coin. Here we propose a rigorous
framework based on the network response over time (i.e., Green function) to
study interactions between nodes across time. For this purpose we define the
\emph{flow} that describes the interplay between the network connectivity and
external inputs. This multivariate measure relates to the concepts of graph
communicability and the map equation. We illustrate our theory using the
multivariate Ornstein-Uhlenbeck process, which describes stable and
non-conservative dynamics, but the formalism can be adapted to other local
dynamics for which the Green function is known. We provide applications to
classical network examples, such as small-world ring and hierarchical networks.
Our theory defines a comprehensive framework that is canonically related to
directed and weighted networks, thus paving a new way to revise the standards
for network analysis, from the pairwise interactions between nodes to the
global properties of networks including community detection.
