Explaining complex network permeability through the covariance of channel length and velocity

We investigate the hydraulic permeability of complex channel networks by expressing analytically the system tortuosity in terms of topology. The tortuosity is shown to be strongly dependent on the covariance between the flow velocity in the channels and their length, and we demonstrate that in isotropic networks no covariance means impermeability. The derived analytical expression for the tortuosity is assessed against numerical simulations of various proximity-based and topology-based isotropic complex networks, and is then used to explain why the permeability of the latter networks is typically lower than the former. The new formula and findings have vast applications in biology, geology, hydrology and in engineering transport systems.