# It’s a Small World for Random Surfers

Article

First Online:

Received:

Accepted:

- 162 Downloads

## Abstract

We prove logarithmic upper bounds for the diameters of the random-surfer Webgraph model and the PageRank-based selection Webgraph model, confirming the small world phenomenon holds for them. In the special case when the generated graph is a tree, we provide close lower and upper bounds for the diameters of both models.

## Keywords

Random-surfer Webgraph model PageRank-based selection model Small-world phenomenon Height of random trees Probabilistic analysis Large deviations## Notes

### Acknowledgments

The authors thank the referees for their careful readings of the manuscript and their many useful comments.

## References

- 1.Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science
**286**(5439), 509–512 (1999)MathSciNetCrossRefMATHGoogle Scholar - 2.Bhamidi, S.: Universal techniques to analyze preferential attachment trees: global and local analysis. (2007). (preprint) http://www.unc.edu/~bhamidi/
- 3.Bianconi, G., Barabási, A.-L.: Competition and multiscaling in evolving networks. Europhys. Lett.
**54**(4), 436–442 (2001)CrossRefGoogle Scholar - 4.Blum, A., Chan, T.-H.H., Rwebangira, M.R.: A random-surfer web-graph model. In: Proceedings of the 8th Workshop on Algorithm Engineering and Experiments and the 3rd Workshop on Analytic Algorithmics and Combinatorics, ALENEX/ANALCO ’06, pp. 238–246 (2006)Google Scholar
- 5.Bollobás, B., Riordan, O.: The diameter of a scale-free random graph. Combinatorica
**24**(1), 5–34 (2004)MathSciNetCrossRefMATHGoogle Scholar - 6.Bonato, A., Chung, F.: Complex networks. In: Gross, J.L., Yellen, J., Zhang, P. (eds.) Handbook of Graph Theory, Chapter 12.1, 2nd edn, pp. 1456–1476. Chapman & Hall, London (2013)Google Scholar
- 7.Broutin, N., Devroye, L.: Large deviations for the weighted height of an extended class of trees. Algorithmica
**46**(3–4), 271–297 (2006)MathSciNetCrossRefMATHGoogle Scholar - 8.Chakrabarti, D., Faloutsos, C.: Graph mining: laws, generators, and algorithms. ACM Comput. Surv.
**38**(1), (2006). Article 2Google Scholar - 9.Chebolu, P., Melsted, P.: Pagerank and the random surfer model. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms. SODA ’08, pp. 1010–1018. Philadelphia, PA, USA (2008)Google Scholar
- 10.Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, Volume 38 of Stochastic Modelling and Applied Probability. Springer, Berlin (2010). Corrected reprint of the second (1998) editionGoogle Scholar
- 11.Devroye, L., Fawzi, O., Fraiman, N.: Depth properties of scaled attachment random recursive trees. Random Struct. Algorithms
**41**(1), 66–98 (2012)MathSciNetCrossRefMATHGoogle Scholar - 12.Dommers, S., van der Hofstad, R., Hooghiemstra, G.: Diameters in preferential attachment models. J. Stat. Phys.
**139**(1), 72–107 (2010)MathSciNetCrossRefMATHGoogle Scholar - 13.Drinea, E., Frieze, A., Mitzenmacher, M.: Balls and bins models with feedback. In: Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms. SODA ’02, pp. 308–315. Philadelphia, PA, USA (2002)Google Scholar
- 14.Ergün, G., Rodgers, G.J.: Growing random networks with fitness. Phys. A Stat. Mech. Appl.
**303**(1–2), 261–272 (2002)CrossRefMATHGoogle Scholar - 15.Krapivsky, P.L., Redner, S.: Organization of growing random networks. Phys. Rev. E
**63**, 066123 (2001)CrossRefGoogle Scholar - 16.Leskovec, J., Kleinberg, J., Faloutsos, C.: Graph evolution: densification and shrinking diameters. ACM Trans. Knowl. Discov. Data (TKDD)
**1**(1), (2007). Article 2Google Scholar - 17.Pandurangan, G., Raghavan, P., Upfal, E.: Using pagerank to characterize web structure. Internet Math.
**3**(1), 1–20 (2006). Conference version in COCOON 2002Google Scholar - 18.Pittel, B.: Note on the heights of random recursive trees and random \(m\)-ary search trees. Random Struct. Algorithms
**5**(2), 337–347 (1994)MathSciNetCrossRefMATHGoogle Scholar

## Copyright information

© Springer Science+Business Media New York 2015