The distribution of path lengths of self avoiding walks on Erd″s-Rényi networks

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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 Erdõs-Rényi networks we show that the pruned networks maintain a Poisson degree distribution, pk t(κ), with an average degree, kt ℓ , that decreases linearly in time. We enumerate the SAW paths of any given length and find that the number of paths, n l T (κ), increases dramatically as a function of l. We also obtain analytical results for the path-length distribution, P l (κ), of the SAW paths which are actually pursued, starting from a random initial node. It turns out that P l (κ) follows the Gompertz distribution, which means that the termination probability of an SAW path increases with its length.

Original languageAmerican English
Article number285002
JournalJournal of Physics A: Mathematical and Theoretical
Volume49
Issue number28
DOIs
StatePublished - 1 Jun 2016

Bibliographical note

Publisher Copyright:
© 2016 IOP Publishing Ltd.

Keywords

  • Erdõs-Rényi network
  • Random network
  • attrition length
  • degree distribution
  • last hitting time
  • random walk
  • self-avoiding walk

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