Uncovering hidden flows in physical networks

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.