  We present an analytical and numerical study of the paths of self avoiding
walks (SAWs) on random networks. Since these walks do not retrace their paths,
they effectively delete the nodes they visit, together with their links, thus
pruning the network. The walkers hop between neighboring nodes, until they
reach a dead-end node from which they cannot proceed. Focusing on
Erd\H{o}s-R\'enyi networks we show that the pruned networks maintain a Poisson
degree distribution, $p_t(k)$, with an average degree, $\langle k \rangle_t$,
that decreases linearly in time. We enumerate the SAW paths of any given length
and find that the number of paths, $n_T(\ell)$, increases dramatically as a
function of $\ell$. We also obtain analytical results for the path-length
distribution, $P(\ell)$, of the SAW paths which are actually pursued, starting
from a random initial node. It turns out that $P(\ell)$ follows the Gompertz
distribution, which means that the termination probability of an SAW path
increases with its length.
