80 research outputs found
Efficient Simulation for Branching Linear Recursions
We consider a linear recursion of the form where is a
real-valued random vector with ,
is a sequence of i.i.d. copies of ,
independent of , and denotes
equality in distribution. For suitable vectors and
provided the initial distribution of is well-behaved, the process
is known to converge to the endogenous solution of the corresponding
stochastic fixed-point equation, which appears in the analysis of information
ranking algorithms, e.g., PageRank, and in the complexity analysis of divide
and conquer algorithms, e.g. Quicksort. Naive Monte Carlo simulation of
based on the branching recursion has exponential complexity in ,
and therefore the need for efficient methods. We propose in this paper an
iterative bootstrap algorithm that has linear complexity and can be used to
approximately sample . We show the consistency of estimators based on
our proposed algorithm.Comment: submitted to WSC 201
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 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 where is a
real-valued vector with , , and the are i.i.d. copies of ,
independent of . Moreover, we
provide precise asymptotics for the limit , which when the
in-degree distribution in the directed configuration model has a power law
imply a power law distribution for with the same exponent
Allocating Divisible Resources on Arms with Unknown and Random Rewards
We consider a decision maker allocating one unit of renewable and divisible
resource in each period on a number of arms. The arms have unknown and random
rewards whose means are proportional to the allocated resource and whose
variances are proportional to an order of the allocated resource. In
particular, if the decision maker allocates resource to arm in a
period, then the reward is, where
is the unknown mean and the noise is independent and
sub-Gaussian. When the order ranges from 0 to 1, the framework smoothly
bridges the standard stochastic multi-armed bandit and online learning with
full feedback. We design two algorithms that attain the optimal gap-dependent
and gap-independent regret bounds for , and demonstrate a phase
transition at . The theoretical results hinge on a novel concentration
inequality we have developed that bounds a linear combination of sub-Gaussian
random variables whose weights are fractional, adapted to the filtration, and
monotonic
Coupling on weighted branching trees
This paper considers linear functions constructed on two different weighted
branching processes and provides explicit bounds for their
Kantorovich-Rubinstein distance in terms of couplings of their corresponding
generic branching vectors. Motivated by applications to the analysis of random
graphs, we also consider a variation of the weighted branching process where
the generic branching vector has a different dependence structure from the
usual one. By applying the bounds to sequences of weighted branching processes,
we derive sufficient conditions for the convergence in the
Kantorovich-Rubinstein distance of linear functions. We focus on the case where
the limits are endogenous fixed points of suitable smoothing transformations
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Ranking Algorithms on Directed Configuration Networks
In recent decades, complex real-world networks, such as social networks, the World Wide Web, financial networks, etc., have become a popular subject for both researchers and practitioners. This is largely due to the advances in computing power and big-data analytics. A key issue of analyzing these networks is the centrality of nodes. Ranking algorithms are designed to achieve the goal, e.g., Google's PageRank. We analyze the asymptotic distribution of the rank of a randomly chosen node, computed by a family of ranking algorithms on a random graph, including PageRank, when the size of the network grows to infinity.
We propose a configuration model generating the structure of a directed graph given in- and out-degree distributions of the nodes. The algorithm guarantees the generated graph to be simple (without self-loops and multiple edges in the same direction) for a broad spectrum of degree distributions, including power-law distributions. Power-law degree distribution is referred to as scale-free property and observed in many real-world networks. On the random graph G_n=(V_n,E_n) generated by the configuration model, we study the distribution of the ranks, which solves
R_i=β _{j: (j,i) β E_n} (C_jR_j +Q_i)
for all node i, some weight C_i and personalization value Q_i.
We show that as the size of the graph n β β, the rank of a randomly chosen node converges weakly to the endogenous solution of the
R =^D β _{i=1}^N (C_iR_i + Q),
where (Q, N, {C_i}) is a random vector and {R_i} are i.i.d. copies of R, independent of (Q, N,{C_i}). This main result is divided into three steps. First, we show that the rank of a randomly chosen node can be approximated by applying the ranking algorithm on the graph for finite iterations. Second, by coupling the graph to a branching tree that is governed by the empirical size-biased distribution, we approximate the finite iteration of the ranking algorithm by the root node of the branching tree. Finally, we prove that the rank of the root of the branching tree converges to that of a limiting weighted branching process, which is independent of n and solves the stochastic fixed-point equation. Our result formalizes the well-known heuristics, that a network often locally possesses a tree-like structure. We conduct a numerical example showing that the approximation is very accurate for English Wikipedia pages (over 5 million).
To draw a sample from the endogenous solution of the stochastic fixed-point equation, one can run linear branching recursions on a weighted branching process. We provide an iterative simulation algorithm based on bootstrap. Compared to the naive Monte Carlo, our algorithm reduces the complexity from exponential to linear in the number of recursions. We show that as the bootstrap sample size tends to infinity, the sample drawn according to our algorithm converges to the target distribution in the Kantorovich-Rubinstein distance and the estimator is consistent
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