The distribution of path lengths of self avoiding walks on ErdősRényi networks
Abstract
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 deadend node from which they cannot proceed. Focusing on ErdősRényi networks we show that the pruned networks maintain a Poisson degree distribution, {p}_{t}(k), with an average degree, < k{> }_{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 pathlength 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.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 July 2016
 DOI:
 10.1088/17518113/49/28/285002
 arXiv:
 arXiv:1603.06613
 Bibcode:
 2016JPhA...49B5002T
 Keywords:

 Condensed Matter  Disordered Systems and Neural Networks;
 Physics  Physics and Society
 EPrint:
 24 pages, 11 figures