arXiv subjects

Mariana Olvera-Cravioto

Publications and source records attributed to Mariana Olvera-Cravioto.

29 records · Page 2Linked to original sources

arXiv2014

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 $\mathcal{R}^*$ 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 $$\mathcal{R} \stackrel{\mathcal{D}}{=} \sum_{i=1}^{\mathcal{N}} \mathcal{C}_i \mathcal{R}_i + \mathcal{Q},$$ where $(\mathcal{Q}, \mathcal{N}, \{ \mathcal{C}_i\})$ is a real-valued vector with $\mathcal{N} \in \{0,1,2,\dots\}$, $P(|\mathcal{Q}| > 0) > 0$, and the $\{\mathcal{R}_i\}$ are i.i.d. copies of $\mathcal{R}$, independent of $(\mathcal{Q}, \mathcal{N}, \{ \mathcal{C}_i\})$. Moreover, we provide precise asymptotics for the limit $\mathcal{R}^*$, which when the in-degree distribution in the directed configuration model has a power law imply a power law distribution for $\mathcal{R}^*$ with the same exponent.

math.PR↗

arXiv2014

PageRank in scale-free random graphs

We analyze the distribution of PageRank on a directed configuration model and show that as the size of the graph grows to infinity it can be closely approximated by the PageRank of the root node of an appropriately constructed tree. This tree approximation is in turn related to the solution of a linear stochastic fixed point equation that has been thoroughly studied in the recent literature.

math.PR↗

arXiv2014

Maximums on Trees

We study the minimal/endogenous solution $R$ to the maximum recursion on weighted branching trees given by $$R\stackrel{\mathcal{D}}{=}\left(\bigvee_{i=1}^NC_iR_i \right)\vee Q,$$ where $(Q,N,C_1,C_2,\dots)$ is a random vector with $N\in \mathbb{N}\cup\{\infty\}$, $P(|Q|>0)>0$ and nonnegative weights $\{C_i\}$, and $\{R_i\}_{i\in\mathbb{N}}$ is a sequence of i.i.d. copies of $R$ independent of $(Q,N,C_1,C_2,\dots)$; $\stackrel{\mathcal{D}}{=}$ denotes equality in distribution. Furthermore, when $Q>0$ this recursion can be transformed into its additive equivalent, which corresponds to the maximum of a branching random walk and is also known as a high-order Lindley equation. We show that, under natural conditions, the asymptotic behavior of $R$ is power-law, i.e., $P(|R|>x)\sim Hx^{-α}$, for some $α>0$ and $H>0$. This has direct implications for the tail behavior of other well known branching recursions.

math.PR↗

arXiv2012

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.

math.PR↗

arXiv2012

Implicit Renewal Theory and Power Tails on Trees

We extend Goldie's (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: R =_D sum_{i=1}^N C_i R_i + Q, R =_D max(max_{i=1}^N C_i R_i, Q), and similar recursions, where (Q, N, C_1,..., C_N) is a nonnegative random vector with N in {0, 1, 2, 3, ..., infinity}, and {R_i}_{i >= 1} are iid copies of R, independent of (Q, N, C_1,..., C_N); =_D denotes the equality in distribution.

math.PR↗

arXiv2012

Asymptotics for Weighted Random Sums

Let $\{X_i\}$ be a sequence of independent identically distributed random variables with an intermediate regularly varying (IR) right tail $\bar{F}$. Let $(N, C_1, ..., C_N)$ be a nonnegative random vector independent of the $\{X_i\}$ with $N \in \mathbb{N} \cup \{\infty\}$. We study the weighted random sum $S_N = \sum_{i=1}^N C_i X_i$, and its maximum, $M_N = \sup_{1 \leq k < N+1} \sum_{i=1}^k C_i X_i$. These type of sums appear in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. In particular, we derive conditions under which $$P(M_N > x) \sim P(S_N > x) \sim E[\sum_{i=1}^N \bar{F}(x/C_i)],$$ as $x \to \infty$. When $E[X_1] > 0$ and the distribution of $Z_N = \sum_{i=1}^N C_i$ is also IR, we obtain the asymptotics $$P(M_N > x) \sim P(S_N > x) \sim E[\sum_{i=1}^N \bar{F}(x/C_i)] + P(Z_N > x/E[X_1]).$$ For completeness, when the distribution of $Z_N$ is IR and heavier than $\bar{F}$, we also obtain conditions under which the asymptotic relations $$P(M_N > x) \sim P(S_N > x) \sim P(Z_N > x/E[X_1])$$ hold.

math.PR↗

arXiv2011

Implicit Renewal Theorem for Trees with General Weights

Consider distributional fixed point equations of the form R =d f(C_i, R_i, 1 <= i <= N), where f(.) is a possibly random real valued function, N in {0, 1, 2, 3,...} U {infty}, {C_i}_{i=1}^N are real valued random weights and {R_i}_{i >= 1} are iid copies of R, independent of (N, C_1,..., C_N); =d represents equality in distribution. Fixed point equations of this type are of utmost importance for solving many applied probability problems, ranging from average case analysis of algorithms to statistical physics. We develop an Implicit Renewal Theorem that enables the characterization of the power tail behavior of the solutions R to many equations of multiplicative nature that fall in this category. This result extends the prior work in Jelenkovic and Olvera-Cravioto (2010), which assumed nonnegative weights {C_i}, to general real valued weights. We illustrate the developed theorem by deriving the power tail asymptotics of the solution R to the linear equation R =d sum_{i=1}^N C_i R_i + Q.

math.PR↗

arXiv2011

Tail behavior of solutions of linear recursions on trees

Consider the linear nonhomogeneous fixed point equation R =_d sum_{i=1}^N C_i R_i + Q, where (Q,N,C_1,...,C_N) is a random vector with N in{0,1,2,3,...}U{infty}, {C_i}_{i=1}^N >= 0, P(|Q|>0) > 0, and {R_i}_{i=1}^N is a sequence of i.i.d. random variables independent of (Q,N,C_1,...,C_N) having the same distribution as R. It is known that R will have a heavy-tailed distribution under several different sets of assumptions on the vector (Q,N,C_1,...,C_N). This paper investigates the settings where either Z_N = sum_{i=1}^N C_i or Q are regularly varying with index -alpha < -1 and E[sum_{i=1}^N C_i^alpha] < 1. This work complements previous results showing that P(R>t) Ht^{-alpha} provided there exists a solution alpha > 0 to the equation E[sum_{i=1}^N|C_i|^alpha] = 1, and both Q and Z_N have lighter tails.

math.PR↗

arXiv2011

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, $W_{\infty}$, of the M/G/1 queue are the so-called heavy-traffic approximation and heavy-tailed asymptotic, respectively. If the traffic intensity, $ρ$, is close to 1 and the processing times have finite variance, the heavy-traffic approximation states that the distribution of $W_{\infty}$ is roughly exponential at scale $O((1-ρ)^{-1})$, while the heavy tailed asymptotic describes power law decay in the tail of the distribution of $W_{\infty}$ 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 $(1-ρ)$, 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.

math.PR↗

arXiv2011

Uniform Approximations for the M/G/1 Queue with Subexponential Processing Times

This paper studies the asymptotic behavior of the steady-state waiting time, W_infty, of the M/G/1 queue with subexponenential processing times for different combinations of traffic intensities and overflow levels. In particular, we provide insights into the regions of large deviations where the so-called heavy traffic approximation and heavy tail asymptotic hold. For queues whose service time distribution decays slower than e^{-sqrt{t}} we identify a third region of asymptotics where neither the heavy traffic nor the heavy tailed approximations are valid. These results are obtained by deriving approximations for P(W_infty > x) that are either uniform in the traffic intensity as the tail value goes to infinity or uniform on the positive axis as the traffic intensity converges to one. Our approach makes clear the connection between the asymptotic behavior of the steady-state waiting time distribution and that of an associated random walk.

math.PR↗

arXiv2010

Information Ranking and Power Laws on Trees

We study the situations when the solution to a weighted stochastic recursion has a power law tail. To this end, we develop two complementary approaches, the first one extends Goldie's (1991) implicit renewal theorem to cover recursions on trees; and the second one is based on a direct sample path large deviations analysis of weighted recursive random sums. We believe that these methods may be of independent interest in the analysis of more general weighted branching processes as well as in the analysis of algorithms.

math.PR↗