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We present an analytical and numerical study of the paths of self avoidingwalks (SAWs) on random networks. Since these walks do not retrace their paths,they effectively delete the nodes they visit, together with their links, thuspruning the network. The walkers hop between neighboring nodes, until theyreach a dead-end node from which they cannot proceed. Focusing onErd\H{o}s-R\'enyi networks we show that the pruned networks maintain a Poissondegree 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 lengthand find that the number of paths, $n_T(\ell)$, increases dramatically as afunction of $\ell$. We also obtain analytical results for the path-lengthdistribution, $P(\ell)$, of the SAW paths which are actually pursued, startingfrom a random initial node. It turns out that $P(\ell)$ follows the Gompertzdistribution, which means that the termination probability of an SAW pathincreases with its length.

The distribution of path lengths of self avoiding walks on Erd\H{o}s-R\'enyi networks