The distribution of path lengths of self avoiding walks on Erdős-Rényi 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 dead-end node from which they cannot proceed. Focusing on Erdős-Rényi 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 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.

24 pages, 11 figures

References in corpus (5)

• Percolation of localized attack on complex networks
• Exploring Complex Networks through Random Walks
• Analytical results for the distribution of shortest path lengths in random networks
• Kinetic-growth self-avoiding walks on small-world networks
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• Analytical results for the distribution of cover times of random walks on random regular graphs
• Self-avoiding walks and connective constants in clustered scale-free networks
• The distribution of first hitting times of non-backtracking random walks on Erdős-Rényi networks
• Efficient network exploration by means of resetting self-avoiding random walkers
• The Compression method and applications
• Analytical results for the distribution of first return times of non-backtracking random walks on configuration model networks
• The effect of preferential node deletion on the structure of networks that evolve via preferential attachment