  In this paper, we prove the existence of fundamental relations between
information theory and estimation theory for network-coded flows. When the
network is represented by a directed graph G=(V, E) and under the assumption of
uncorrelated noise over information flows between the directed links connecting
transmitters, switches (relays), and receivers. We unveil that there yet exist
closed-form relations for the gradient of the mutual information with respect
to different components of the system matrix M. On the one hand, this result
opens a new class of problems casting further insights into effects of the
network topology, topological changes when nodes are mobile, and the impact of
errors and delays in certain links into the network capacity which can be
further studied in scenarios where one source multi-sinks multicasts and
multi-source multicast where the invertibility and the rank of matrix M plays a
significant role in the decoding process and therefore, on the network
capacity. On the other hand, it opens further research questions of finding
precoding solutions adapted to the network level.
