Skip to main content
SpringerLink
Log in

Generating Functions of Waiting Times and Numbers of Visits for Random Walks on Graphs

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

In this paper, we consider some cover time problems for random walks on graphs in a wide class of waiting time problems. By using generating functions, we present a unified approach for the study of distributions associated with waiting times. In addition, the distributions of the numbers of visits for the random walks on the graphs are also studied. We present the relationship between the distributions of the waiting times and the numbers of visits. We also show that these theoretical results can be easily carried out through some computer algebra systems and present some numerical results for cover times in order to demonstrate the usefulness of the results developed. Finally, the study of cover time problems through generating functions leads to more extensive development.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bingham NH (2001) Random walk and fluctuation theory. In: Shanbhag DN, Rao CR (eds) Handbook of statistics, vol 19. Elsevier, Amsterdam, pp 171–213

    Google Scholar 

  • Blom G, Sandell D (1992) Cover times for random walks on graphs. Math Sci 17:111–119

    MathSciNet  MATH  Google Scholar 

  • Chen HY (2007) The generating functions of hitting times for random walk on trees. Stat Probab Lett 77:1574–1579

    Article  MATH  Google Scholar 

  • Doyle PG, Snell JL (1984) Random walks and electrical networks. The Mathematical Association of America, Washington, DC

    Google Scholar 

  • Feller W (1968) An introduction to probability theory and its applications, vol I, 3rd edn. Wiley, New York

    Google Scholar 

  • Graham RL, Knuth DE, Patashnik O (1994) Concrete mathematics, 2nd edn. Addison-Wesley, Reading

    MATH  Google Scholar 

  • Palacios JL (2009) On hitting times of random walks on trees. Stat Probab Lett 79:234–236

    Article  MathSciNet  MATH  Google Scholar 

  • Palacios JL, Renom JM, Berrizbeitia P (1999) Random walks on edge transitive graphs (II). Stat Probab Lett 43:25–32

    Article  MathSciNet  MATH  Google Scholar 

  • Spitzer FL (2001) Principle of random walk, 2nd edn. Springer, New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Economics, Seikei University, 3-3-1 Kichijoji-Kitamachi, Musasino-shi, Tokyo, 180-8633, Japan

    Kiyoshi Inoue

  2. Department of Mathematics, Faculty of Engineering Science, Kansai University, 3-3-35 Yamate-cho, Suita-shi, Osaka, 564-8680, Japan

    Sigeo Aki

  3. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada

    Balakrishnan Narayanaswamy

Authors
  1. Kiyoshi Inoue
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sigeo Aki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Balakrishnan Narayanaswamy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kiyoshi Inoue.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Inoue, K., Aki, S. & Narayanaswamy, B. Generating Functions of Waiting Times and Numbers of Visits for Random Walks on Graphs. Methodol Comput Appl Probab 15, 349–362 (2013). https://doi.org/10.1007/s11009-011-9246-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-011-9246-7

Keywords

AMS 2000 Subject Classifications

Search

Navigation