Subdominant eigenvalues for stochastic matrices with given column sums

Kirkland, Steve

doi:10.13001/1081-3810.1345

Cited by 15 publications

(10 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By the proof of Proposition 4.1, we have that ρ , is equivalent to the last of Inequality (10), that is,…”

Section: Special Choices Of α I For the Set γ Stol α (A)mentioning

confidence: 91%

See 1 more Smart Citation

A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices

Wang

2018

Open Mathematics

View full text Add to dashboard Cite

A set in the complex plane which involves n parameters in [ , ] is given to localize all eigenvalues di erent from 1 for stochastic matrices. As an application of this set, an upper bound for the moduli of the subdominant eigenvalues of a stochastic matrix is obtained. Lastly, we x n parameters in [ , ] to give a new set including all eigenvalues di erent from , which is tighter than those provided by Shen et al.

show abstract

“…By the proof of Proposition 4.1, we have that ρ , is equivalent to the last of Inequality (10), that is,…”

Section: Special Choices Of α I For the Set γ Stol α (A)mentioning

confidence: 91%

“…From the Perron-Frobenius Theorem [7], for any eigenvalue λ of A, that is, λ ∈ σ(A), we have λ ≤ [8]. Here we call λ a moduli of subdominant eigenvalue of a stochastic matrix A if > λ > η for every eigenvalue η di erent from and λ [8][9][10].…”

Section: Introductionmentioning

confidence: 99%

A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices

Wang

2018

Open Mathematics

View full text Add to dashboard Cite

show abstract

“…For Markov matrices, not only does λ = 1 exist but it is also the largest eigenvalue in magnitude. 17 Thus, the principal eigenvector of M can be used to give the steady-state behavior of the system. This derivation is applicable to this model because the representative transition matrices developed by the MDP (Step 2 above) are Markov chain transition matrices.…”

Section: Methodsmentioning

confidence: 99%

An early-stage design model for estimating ship evacuation patterns using the ship-centric Markov decision process

Kana

Droste

2017

Proceedings of the Institution of Mechanical Engineers, Part M:

View full text Add to dashboard Cite

An early-stage design model is presented that estimates personnel locations on board a vessel during times of evacuation. This model takes into account various levels of uncertainty and pain that individuals may feel while heading toward safety, while simultaneously not requiring highly detailed information regarding the vessel layout. This makes this model suitable for analysis during early stages of design. To do this, principal eigenvector analysis is applied to the ship-centric Markov decision process model. Principal eigenvector analysis provides a leading indicator metric for forecasting and quantifying locations of individuals when coupled with the ship-centric Markov decision process model. For evacuation models suited for later stages of design, full temporal simulations may be required to understand long-term implications of personnel movement. This article proposes an alternative method that is able to identify some of these implications while not requiring full details of the vessel layout nor temporal simulations. To do this, a common theorem in Markov theory is applied that defines how the principal eigenvector represents the long-term steady-state behavior of the system. Metrics are defined that quantify the probability that an individual will congregate at specific locations on the vessels and highlight sensitivities to long-term behavior. A case study of a simplified vessel layout is presented that examines decision-making regarding ship egress analysis and general arrangements design. The results highlight specific areas of interest that cause significant changes to where individuals congregate and the probability they arrive safely at the exit. Sensitivity studies are performed varying the uncertainty in the movement of the individuals, how much pain they are experiencing, and one example where a passageway is blocked.

show abstract

“…Let G denote the induced graph of A. To prove that λ 2 (A) < 1, it suffices to show that G has exactly one initial class and that this class is aperiodic [31]. Since A is stochastic and…”

Section: B Two Extreme Cases: Isolation and Fusionmentioning

confidence: 99%

Distributed Inference Over Directed Networks: Performance Limits and Optimal Design

Bajović

Moura

Sinopoli

2016

IEEE Trans. Signal Process.

View full text Add to dashboard Cite

We find large deviations rates for consensus-based distributed inference for directed networks. When the topology is deterministic, we establish the large deviations principle and find exactly the corresponding rate function, equal at all nodes. We show that the dependence of the rate function on the stochastic weight matrix associated with the network is fully captured by its left eigenvector corresponding to the unit eigenvalue. Further, when the sensors' observations are Gaussian, the rate function admits a closed form expression. Motivated by these observations, we formulate the optimal network design problem of finding the left eigenvector which achieves the highest value of the rate function, for a given target accuracy. This eigenvector therefore minimizes the time that the inference algorithm needs to reach the desired accuracy. For Gaussian observations, we show that the network design problem can be formulated as a semidefinite (convex) program, and hence can be solved efficiently. When observations are identically distributed across agents, the system exhibits an interesting property: the graph of the rate function always lies between the graphs of the rate function of an isolated node and the rate function of a fusion center that has access to all observations. We prove that this fundamental property holds even when the topology and the associated system matrices change randomly over time, with arbitrary distribution. Due to generality of its assumptions, the latter result requires more subtle techniques than the standard large deviations tools, contributing to the general theory of large deviations.

show abstract

Subdominant eigenvalues for stochastic matrices with given column sums

Cited by 15 publications

References 8 publications

A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices

A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices

An early-stage design model for estimating ship evacuation patterns using the ship-centric Markov decision process

Distributed Inference Over Directed Networks: Performance Limits and Optimal Design

Contact Info

Product

Resources

About