  Many situations in physics, biology, and engineering consist of the transport
of some physical quantity through a network of narrow channels. The ability of
a network to transport such a quantity in every direction can be described by
the average conductivity associated with. When the flow through each channel is
conserved and derives from a potential function, we show that there exist an
upper bound of the average conductivity, and explicitly give the expression for
this upper bound as a function of the channel permeability and channel length
distributions. Moreover, we express the necessary and sufficient conditions on
the network structure to maximize the average conductivity. These conditions
are found to be independent of the connectivity of the vertices.
