  Understanding the interactions among nodes in a complex network is of great
importance, since they disclose how these nodes are cooperatively supporting
the functioning of the network. Scientists have developed numerous methods to
uncover the underlying adjacent physical connectivity based on measurements of
functional quantities of the nodes states. Often, the physical connectivity,
the adjacency matrix, is available. Yet, little is known about how this
adjacent connectivity impacts on the "hidden" flows being exchanged between any
two arbitrary nodes, after travelling longer non-adjacent paths. In this
Letter, we show that hidden physical flows in conservative flow networks, a
quantity that is usually inaccessible to measurements, can be determined by the
interchange of physical flows between any pair of adjacent nodes. Our approach
applies to steady or dynamic state of either linear or non-linear complex
networks that can be modelled by conservative flow networks, such as gas supply
networks, water supply networks and power grids.
