Information ranking and power laws on trees

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 R d = Q + N i=1 C i R i , 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 R; ' d =' stands for equality in distribution. We study the asymp-totic 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 E N E[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, E N E[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.

Power Laws on Weighted Branching Trees

“…In particular, if the {C i } are iid and independent of N, the condition reduces to E[N α ] < ∞ since E[C α ] < ∞ is implied by the other conditions; see Theorems 4.2 and 5.4 in [26]. Furthermore, when 0…”

“…. ) having the same distribution as R. This recursion has been proposed as a stochastic approximation of Google's PageRank algorithm and possibly other ranking schemes of large information sets, e.g., the World Wide Web (WWW); see [46,47,26] and the references therein. There it is argued that the stochastic approach is helpful in understanding the qualitative behavior of PageRank given the large scale nature of the WWW.…”

“…This stochastic setup was first introduced in [46], where the process resulting after a finite number of iterations of (3) was analyzed. Further generalization of (3) leads to (1), which was recently analyzed in [47,26]. Furthermore, in computer science, a well known divide-and-conquer paradigm is used for designing efficient algorithms, where a problem is recursively divided into two or more sub-problems, until the sub-problems become simple enough to be solved directly.…”

“…In this paper we present some of our recent work from [26,27,28,36] that studies the power-tail asymptotics of the solution R to the preceding distributional fixed-point equations; all the omitted proofs can be found in these references. We will exemplify our techniques primarily on the nonhomogeneous equation (1), for which the tail of the solution, P(R > t), can be determined by three different factors: the multiplicative effect of the weights C i ; the sum of the weights ∑Ci; and the innovation variable Q.…”

“…Fixed-point equations of this type are important for solving many applied probability problems, ranging from the average case analysis of algorithms to statistical physics. In this paper we present some of our recent work from [26,27,28,36] that studies the power tail asymptotics of such solutions. We exemplify our techniques primarily on the nonhomogeneous equation, R D = ∑ N i=1 C i R i + Q, for which the power tail of the solution, P(R > t), can be determined by three different factors: the multiplicative effect of the weights C i ; the sum of the weights ∑Ci; and the innovation variable Q.…”