On Approximating the Stationary Distribution of Time-Reversible Markov Chains

Bressan, Marco; Peserico, Enoch; Pretto, Luca

doi:10.1007/s00224-019-09921-3

Cited by 8 publications

(6 citation statements)

References 20 publications

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“…Most related is the work of [30], which also try to identify the most relevant parts of the system, however they employ the special structure given by cellu-lar processes to find these regions and estimate the subsequent approximation error. Many other works deal with special cases, such as queueing models [1,18], time-reversible chains [9], or positive rows (all states have a transition to one particular state) [10,12,27]. In contrast, our methods aim to deal with general Markov chains.…”

Section: Related Workmentioning

confidence: 99%

Correct Approximation of Stationary Distributions

Meggendorfer¹

2023

Preprint

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A classical problem for Markov chains is determining their stationary (or steady-state) distribution. This problem has an equally classical solution based on eigenvectors and linear equation systems. However, this approach does not scale to large instances, and iterative solutions are desirable. It turns out that a naive approach, as used by current model checkers, may yield completely wrong results. We present a new approach, which utilizes recent advances in partial exploration and mean payoff computation to obtain a correct, converging approximation.

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Section: Related Workmentioning

confidence: 99%

Correct Approximation of Stationary Distributions

Meggendorfer¹

2023

Preprint

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“…Finally, we would mention recent work on the local approximation of the stationary probability of a target state v in a Markov Chain [47], [50], [51], and on the local approximation of a single entry of the solution vector of a linear system [52], [53]. The local approximation of P (v) is a specific but nontrivial case of both, and we hope that our techniques may serve as an entry point for future developments in those directions.…”

Section: Related Workmentioning

confidence: 99%

Sublinear Algorithms for Local Graph-Centrality Estimation

Bressan,

Peserico,

Pretto

2023

SIAM J. Comput.

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We study the complexity of local graph centrality estimation, with the goal of approximating the centrality score of a given target node while exploring only a sublinear number of nodes/arcs of the graph and performing a sublinear number of elementary operations. We develop a technique, that we apply to the PageRank and Heat Kernel centralities, for building a low-variance score estimator through a local exploration of the graph. We obtain an algorithm that, given any node in any graph of m arcs, with probability (1 − δ) computes a multiplicative (1 ± )-approximation of its score by examining only Õ min m 2/3 Δ 1/3 d −2/3 , m 4/5 d −3/5 nodes/arcs, where Δ and d are respectively the maximum and average outdegree of the graph (omitting for readability poly( −1 ) and polylog(δ −1 ) factors). A similar bound holds for computational cost. We also prove a lower bound of Ω min m 1/2 Δ 1/2 d −1/2 , m 2/3 d −1/3 for both query complexity and computational complexity. Moreover, our technique yields a Õ(n 2/3 )-queries algorithm for an n-node graph in the access model of Brautbar et al. [1], widely used in social network mining; we show this algorithm is optimal up to a sublogarithmic factor. These are the first algorithms yielding worst-case sublinear bounds for general directed graphs and any choice of the target node.

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“…The stationary distribution. Lee et al [23] and Bressan et al [8] studied the question of computing the stationary distribution 𝜋 of a Markov Chain locally. These algorithms take as input any state 𝑣, and answer if the stationary probability of 𝑣 exceeds some Δ ∈ (0, 1) and/or output an estimate of 𝜋 (𝑣).…”

Section: Related Workmentioning

confidence: 99%

Local Algorithms for Estimating Effective Resistance

Pan¹,

Lopatta²,

Yoshida³

et al. 2021

Preprint

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Effective resistance is an important metric that measures the similarity of two vertices in a graph. It has found applications in graph clustering, recommendation systems and network reliability, among others. In spite of the importance of the effective resistances, we still lack efficient algorithms to exactly compute or approximate them on massive graphs.In this work, we design several local algorithms for estimating effective resistances, which are algorithms that only read a small portion of the input while still having provable performance guarantees. To illustrate, our main algorithm approximates the effective resistance between any vertex pair 𝑠, 𝑡 with an arbitrarily small additive error 𝜀 in time 𝑂 (poly(log 𝑛/𝜀)), whenever the underlying graph has bounded mixing time. We perform an extensive empirical study on several benchmark datasets, validating the performance of our algorithms. CCS CONCEPTS• Mathematics of computing → Graph algorithms; • Theory of computation → Sketching and sampling.

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On Approximating the Stationary Distribution of Time-Reversible Markov Chains

Cited by 8 publications

References 20 publications

Correct Approximation of Stationary Distributions

Correct Approximation of Stationary Distributions

Sublinear Algorithms for Local Graph-Centrality Estimation

Local Algorithms for Estimating Effective Resistance

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