Skip to main content Accessibility help
×
Home

Information ranking and power laws on trees

  • Predrag R. Jelenković (a1) and Mariana Olvera-Cravioto (a1)

Abstract

In this paper we consider the stochastic analysis of information ranking algorithms of large interconnected data sets, e.g. Google's PageRank algorithm for ranking pages on the World Wide Web. The stochastic formulation of the problem results in an equation of the form where N, Q, {R i } i≥1, and {C, C i } i≥1 are independent nonnegative random variables, the {C, C i } i≥1 are identically distributed, and the {R i } i≥1 are independent copies of stands for equality in distribution. We study the asymptotic properties of the distribution of R that, in the context of PageRank, represents the frequencies of highly ranked pages. The preceding equation is interesting in its own right since it belongs to a more general class of weighted branching processes that have been found to be useful in the analysis of many other algorithms. Our first main result shows that if ENE[C α] = 1, α > 0, and Q, N satisfy additional moment conditions, then R has a power law distribution of index α. This result is obtained using a new approach based on an extension of Goldie's (1991) implicit renewal theorem. Furthermore, when N is regularly varying of index α > 1, ENE[C α] < 1, and Q, C have higher moments than α, then the distributions of R and N are tail equivalent. The latter result is derived via a novel sample path large deviation method for recursive random sums. Similarly, we characterize the situation when the distribution of R is determined by the tail of Q. The preceding approaches may be of independent interest, as they can be used for analyzing other functionals on trees. We also briefly discuss the engineering implications of our results.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Information ranking and power laws on trees
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Information ranking and power laws on trees
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Information ranking and power laws on trees
      Available formats
      ×

Copyright

Corresponding author

Postal address: Department of Electrical Engineering, Columbia University, New York, NY 10027, USA.
∗∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA. Email address: molvera@ieor.columbia.edu

References

Hide All
[1] Alsmeyer, G. and Kuhlbusch, D. (2010). Double martingale structure and existence of ϕ-moments for weighted branching processes. To appear in Münster J. Math.
[2] Alsmeyer, G. and Rösler, U. (2006). A stochastic fixed point equation related to weighted branching with deterministic weights. Electron. J. Prob. 11, 2756.
[3] Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Prob. 8, 354474.
[4] Asmussen, S. (2003). Applied Probability and Queues. Springer, New York.
[5] Athreya, K. B., McDonald, D. and Ney, P. (1978). Limit theorems for semi-Markov processes and renewal theory for Markov chains. Ann. Prob. 6, 788797.
[6] Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover, Mineola, NY.
[7] Baltrūnas, A., Daley, D. J. and Klüppelberg, C. (2004). Tail behavior of the busy period of a GI/GI/1 queue with subexponential service times. Stoch. Process. Appl. 111, 237258.
[8] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.
[9] Borovkov, A. (2000). Estimates for the distribution of sums and maxima of sums of random variables without the Cramér condition. Siberian Math J. 41, 811848.
[10] Brandt, A. (1986). The stochastic equation y n+1 = a n y n + b n with stationary coefficients. Adv. Appl. Prob. 18, 211220.
[11] Brin, S. and Page, L. (1998). The anatomy of a large-scale hypertextual Web search engine. Comput. Networks ISDN Systems 30, 107117.
[12] Chow, Y. S. and Teicher, H. (1988). Probability Theory, 2nd edn. Springer, New York.
[13] De Meyer, A. and Teugels, J. L. (1980). On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1. J. Appl. Prob. 17, 802813.
[14] Denisov, D., Foss, S. and Korshunov, D. (2009). Asymptotics of randomly stopped sums in the presence of heavy tails. Preprint. Available at the http://arxiv.org/abs/0808.3697v3.
[15] Fill, J. A. and Janson, S. (2001). Approximating the limiting Quicksort distribution. Random Structures Algorithms 19, 376406.
[16] Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126166.
[17] Gyöngyi, Z., Garcia-Molina, H. and Pedersen, J. (2004). Combating Web spam with TrustRank. Tech. Rep., Stanford University.
[18] Iksanov, A. M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Process. Appl. 114, 2750.
[19] Jelenković, P. R. and Momčilović, P. (2004). Large deviations of square root insensitive random sums. Math. Operat. Res. 29, 398406.
[20] Jelenković, P. R. and Olvera-Cravioto, M. (2009). Information ranking and power laws on trees. Preprint. Available at http://arxiv.org/abs/0905.1738.
[21] Jelenković, P. R. and Tan, J. (2010). Modulated branching processes, origins of power laws and queueing duality. Math. Operat. Res. 35, 807829.
[22] Jessen, A. H. and Mikosch, T. (2006). Regularly varying functions. Publ. Inst. Math. 80, 171192.
[23] Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207248.
[24] Kleinberg, J. M. (1999). Authoritative sources in a hyperlinked environment. J. ACM 46, 604632.
[25] Kuhlbusch, D. (2004). On weighted branching processes in random environment. Stoch. Process. Appl. 109, 113144.
[26] Litvak, N., Scheinhardt, W. R. W. and Volkovich, Y. (2007). In-degree and PageRank: why do they follow similar power laws? Internet Math. 4, 175198.
[27] Liu, Q. (1998). Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. Appl. Prob. 30, 85112.
[28] Liu, Q. (2000). On generalized multiplicative cascades. Stoch. Process. Appl. 86, 263286.
[29] Mikosch, T. and Samorodnitsky, G. (2000). The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Prob. 10, 10251064.
[30] Nagaev, S. V. (1982). On the asymptotic behavior of one-sided large deviation probabilities. Theory Prob. Appl. 26, 362366.
[31] Rösler, U. (1993). The weighted branching process. In Dynamics of Complex and Irregular Systems (Bielefeld, 1991), World Science Publishing, River Edge, NJ, pp. 154165.
[32] Rösler, U. and Rüschendorf, L. (2001). The contraction method for recursive algorithms. Algorithmica 29, 333.
[33] Rösler, U., Topchi, V. A. and Vatutin, V. A. (2000). Convergence conditions for the weighted branching process. Discrete Math. Appl. 10, 521.
[34] Volkovich, Y. (2009). Stochastic analysis of Web page ranking. , University of Twente.
[35] Volkovich, Y. and Litvak, N. (2010). Asymptotic analysis for personalized Web search. Adv. Appl. Prob. 42, 577604.
[36] Volkovich, Y., Litvak, N. and Donato, D. (2007). Determining factors behind the Pagerank log-log plot. In Algorithms and Models for the Web-Graph, Springer, Berlin, pp. 108123.
[37] Zwart, A. P. (2001). Tail asymptotics for the busy period in the GI/G/1 queue. Math. Operat. Res. 26, 485493.

Keywords

MSC classification

Related content

Powered by UNSILO

Information ranking and power laws on trees

  • Predrag R. Jelenković (a1) and Mariana Olvera-Cravioto (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.