Analytical results for the distribution of first hitting times of random walks on random regular graphs

Ido Tishby, Ofer Biham, Eytan Katzav*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

We present analytical results for the distribution of first hitting (FH) times of random walks (RWs) on random regular graphs (RRGs) of degree c ≥ 3 and a finite size N. Starting from a random initial node at time t = 1, at each time step t ≥ 2 an RW hops randomly into one of the c neighbors of its previous node. In some of the time steps the RW may hop into a yet-unvisited node while in other time steps it may revisit a node that has already been visited before. The first time at which the RW enters a node that has already been visited before is called the FH time or the first intersection length. The FH event may take place either by backtracking (BT) to the previous node or by retracing (RET), namely stepping into a node which has been visited two or more time steps earlier. We calculate the tail distribution P(TFH > t) of FH times as well as its mean 〈TFH〉 and variance Var(TFH). We also calculate the probabilities PBT and PRET that the FH event will occur via the BT scenario or via the RET scenario, respectively. We show that in dilute networks the dominant FH scenario is BT while in dense networks the dominant scenario is RET and calculate the conditional distributions P(TFH = t|BT) and P(TFH = t|RET), for the two scenarios. The analytical results are in excellent agreement with the results obtained from computer simulations. Considering the FH event as a termination mechanism of the RW trajectories, these results provide useful insight into the general problem of survival analysis and the statistics of mortality rates when two or more termination scenarios coexist.

Original languageAmerican English
Article number145002
JournalJournal of Physics A: Mathematical and Theoretical
Volume54
Issue number14
DOIs
StatePublished - 9 Apr 2021

Bibliographical note

Funding Information:
This work was supported by the Israel Science Foundation Grant No. 1682/18.

Publisher Copyright:
© 2021 IOP Publishing Ltd Printed in the UK

Keywords

  • Backtracking
  • First hitting time
  • First intersection length
  • Random network
  • Random regular graph
  • Random walk
  • Retracing

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