  In this paper we find new scaling laws for the evolution of $p$-brane
networks in $N+1$-dimensional Friedmann-Robertson-Walker universes in the
weakly-interacting limit, giving particular emphasis to the case of cosmic
superstrings ($p=1$) living in a universe with three spatial dimensions (N=3).
In particular, we show that, during the radiation era, the root-mean-square
velocity is ${\bar v} =1/{\sqrt 2}$ and the characteristic length of
non-interacting cosmic string networks scales as $L \propto a^{3/2}$ ($a$ is
the scale factor), thus leading to string domination even when gravitational
backreaction is taken into account. We demonstrate, however, that a small
non-vanishing constant loop chopping efficiency parameter $\tilde c$ leads to a
linear scaling solution with constant $L H \ll 1$ ($H$ is the Hubble parameter)
and ${\bar v} \sim 1/{\sqrt 2}$ in the radiation era, which may allow for a
cosmologically relevant cosmic string role even in the case of light strings.
We also determine the impact that the radiation-matter transition has on the
dynamics of weakly interacting cosmic superstring networks.
