Mariana Olvera-Cravioto

Implicit Renewal Theory and Power Tails on Trees

We extend Goldies (1991) implicit renewal theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power-tail asymptotics of the distributions of the solutions R to and similar recursions, where (Q, N, C 1, C 2,…) is a nonnegative random vector with N ∈ {0, 1, 2, 3,…} ∪ {∞}, and are independent and identically distributed copies of R, independent of (Q, N, C 1, C 2,…); here ‘∨’ denotes the maximum operator.

INFORMATION RANKING AND POWER LAWS ON TREES

In this paper we consider the stochastic analysis of information ranking algorithms of large interconnected data sets, e.g. Googles 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 Goldies (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.

Directed Random Graphs with given Degree Distributions

Given two distributions F and G on the nonnegative integers we propose an algorithm to construct in- and out-degree sequences from samples of i.i.d. observations from F and G, respectively, that with high probability will be graphical, that is, from which a simple directed graph can be drawn. We then analyze a directed version of the configuration model and show that, provided that F and G have finite variance, the probability of obtaining a simple graph is bounded away from zero as the number of nodes grows. We show that conditional on the resulting graph being simple, the in- and out-degree distributions are (approximately) F and G for large size graphs. Moreover, when the degree distributions have only finite mean we show that the elimination of self-loops and multiple edges does not significantly change the degree distributions in the resulting simple graph.

Asymptotics for Weighted Random Sums

Let {X i} be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail F̄. Let (N, C 1, C 2,…) be a nonnegative random vector independent of the {X i } with N∈ℕ∪ {∞}. We study the weighted random sum S N =∑{i=1} N C i X i , and its maximum, M N =sup{1≤k N+1∑ i=1 k C i X i . This type of sum appears in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. In particular, we derive conditions under which P(M N > x)∼ P(S N > x)∼ E[∑ i=1 N F̄(x/C i )] as x→∞. When E[X 1]>0 and the distribution of Z N =∑ i=1 N C i is also intermediate regularly varying, we obtain the asymptotics P(M N > x)∼ P(S N > x)∼ E[∑ i=1 N F̄}(x/C i )] +P(Z N > x/E[X 1]). For completeness, when the distribution of Z N is intermediate regularly varying and heavier than F̄, we also obtain conditions under which the asymptotic relations P(M N > x) ∼ P(S N > x)∼ P(Z N > x / E[X 1 ] hold.

On the transition from heavy traffic to heavy tails for the M/G/1 queue: The regularly varying case.

Two of the most popular approximations for the distribution of the steady-state waiting time, W∞, of the M/G/1 queue are the socalled heavy-traffic approximation and heavy-tailed asymptotic, respectively. If the traffic intensity,�, is close to 1 and the processing times have finite variance, the heavy-traffic approximation s that the distribution of W∞ is roughly exponential at scale O((1 �) −1 ), while the heavy tailed asymptotic describes power law decay in the tail of the distribution of W∞ for a fixed traffic intensity. In this paper, we assume a regularly varying processing time distribution and obtain a sharp threshold in terms of the tail value, or equivalently in terms of (1 �), that describes the point at which the tail behavior transitions from the heavy-traffic regime to the heavy-tailed asymptotic. We also provide new approximations that are either uniform in the traffic intensity, or uniform on the positive axis, that avoid the need to use different expressions on the two regions defined by the threshold. 1. Introduction. A substantial literature has been developed over the last forty years that recognizes the simplifications that arise in the analysis of queueing systems in the presence of “heavy traffic.” The earliest such “heavy traffic” approximation was that obtained byKingman (1961, 1962) for the steady-state waiting time W∞ for the G/G/1 queue. In particular, let Wn be the waiting time (exclusive of service) of the nth customer for a first-in first-out (FIFO) single-server queue (with an infinite capacity waiting room) fed by a renewal arrival process [with i.i.d. inter-arrival times (�n:n ≥ 1)] and an independent stream of i.i.d. processing times (Vn:n ≥ 0). If � ,

Efficient simulation for branching linear recursions

We provide an algorithm for simulating the unique attracting fixed-point of linear branching distributional equations. Such equations appear in the analysis of information ranking algorithms, e.g., PageRank, and in the complexity analysis of divide and conquer algorithms, e.g., Quicksort. The naive simulation approach would be to simulate exactly a suitable number of generations of a weighted branching process, which has exponential complexity in the number of generations being sampled. Instead, we propose an iterative bootstrap algorithm that has linear complexity; we prove its convergence and the consistency of a family of estimators based on our approach.

Ranking algorithms on directed configuration networks

This paper studies the distribution of a family of rankings, which includes Google’s PageRank, on a directed configuration model. In particular, it is shown that the distribution of the rank of a randomly chosen node in the graph converges in distribution to a finite random variable

PageRank in Scale-Free Random Graphs

R^*

Uniform approximations for the M/G/1 queue with subexponential processing times

that can be written as a linear combination of i.i.d. copies of the endogenous solution to a stochastic fixed point equation of the form

Coupling on weighted branching trees

R \stackrel {D}{=} \sum^N _{i=1} C_iR_i + Q,