Inverse problems for random walks on trees: Network tomography

Pena, Victor de la; Gzyl, Henryk; McDonald, Patrick

doi:10.1016/j.spl.2008.06.001

Cited by 3 publications

(2 citation statements)

References 5 publications

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“…While it is easy to see that there can be no general higher-dimensional analog of our results, we have obtained completely analogous results for finite trees [3]. We expect that these results will have applications involving communication networks.…”

Section: P Lm )supporting

confidence: 66%

Hitting Time and Inverse Problems for Markov Chains

Pena

Gzyl

McDonald

2008

Journal of Applied Probability

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Let Wn be a simple Markov chain on the integers. Suppose that Xn is a simple Markov chain on the integers whose transition probabilities coincide with those of Wn off a finite set. We prove that there is an M > 0 such that the Markov chain Wn and the joint distributions of the first hitting time and first hitting place of Xn started at the origin for the sets {-M, M} and {-(M + 1), (M + 1)} algorithmically determine the transition probabilities of Xn.

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Section: P Lm )supporting

confidence: 66%

Hitting Time and Inverse Problems for Markov Chains

Pena

Gzyl

McDonald

2008

Journal of Applied Probability

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“…One method of recovering the parameters is to observe random walks on the network as a weighted graph and try to recover the transition matrix, as the transition probabilities are often directly related to the weights. The inverse problems for random walks were studied in [47,48,76,79], and applications were considered in optical tomography [54,55,56,57,78], network tomography [58,86], electrical resistor networks [39,65,74] and neuroscience [8]. These works use different types of random walk measurements at accessible nodes to recover the transition matrix in different settings, assuming the topology of the network is known.…”

Section: Introductionmentioning

confidence: 99%

Inverse problems for discrete heat equations and random walks for a class of graphs

Blåsten¹,

Isozaki²,

Lassas³

et al. 2021

Preprint

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We study the inverse problem of determining a finite weighted graph (X, E) from the source-to-solution map on a vertex subset B ⊂ X for heat equations on graphs, where the time variable can be either discrete or continuous. We prove that this problem is equivalent to the discrete version of the inverse interior spectral problem, provided that there does not exist a nonzero eigenfunction of the weighted graph Laplacian vanishing identically on B. In particular, we consider inverse problems for discrete-time random walks on finite graphs. We show that under the Two-Points Condition, the graph structure and the transition matrix of the random walk can be uniquely recovered from the distributions of the first passing times on B, or from the observation on B of one realization of the random walk.

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