  We define gradient networks as directed graphs formed by local gradients of a
scalar field distributed on the nodes of a substrate network G. We derive an
exact expression for the in-degree distribution of the gradient network when
the substrate is a binomial (Erdos-Renyi) random graph, G(N,p). Using this
expression we show that the in-degree distribution R(l) of gradient graphs on
G(N,p) obeys the power law R(l)~1/l for arbitrary, i.i.d. random scalar fields.
We then relate gradient graphs to congestion tendency in network flows and show
that while random graphs become maximally congested in the large network size
limit, scale-free networks are not, forming fairly efficient substrates for
transport. Combining this with other constraints, such as uniform edge cost, we
obtain a plausible argument in form of a selection principle, for why a number
of spontaneously evolved massive networks are scale-free. This paper also
presents detailed derivations of the results recently reported in Nature, vol.
428, pp. 716 (2004).
