A Method to Calculate Steady-State Distributions of Large Markov Chains by Aggregating States

1987 ◽  
Vol 35 (2) ◽  
pp. 282-290 ◽  
Author(s):  
Brion N. Feinberg ◽  
Samuel S. Chiu
Keyword(s):  
Steady State ◽  
Markov Chains

scholarly journals Steady-State Analysis of Genetic Regulatory Networks Modelled by Probabilistic Boolean Networks

2003 ◽  
Vol 4 (6) ◽  
pp. 601-608 ◽  
Author(s):  
Ilya Shmulevich ◽  
Ilya Gluhovsky ◽  
Ronaldo F. Hashimoto ◽  
Edward R. Dougherty ◽  
Wei Zhang
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Markov Chains ◽  
Regulatory Networks ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
Long Run ◽  
Probabilistic Boolean Networks

Probabilistic Boolean networks (PBNs) have recently been introduced as a promising class of models of genetic regulatory networks. The dynamic behaviour of PBNs can be analysed in the context of Markov chains. A key goal is the determination of the steady-state (long-run) behaviour of a PBN by analysing the corresponding Markov chain. This allows one to compute the long-term influence of a gene on another gene or determine the long-term joint probabilistic behaviour of a few selected genes. Because matrix-based methods quickly become prohibitive for large sizes of networks, we propose the use of Monte Carlo methods. However, the rate of convergence to the stationary distribution becomes a central issue. We discuss several approaches for determining the number of iterations necessary to achieve convergence of the Markov chain corresponding to a PBN. Using a recently introduced method based on the theory of two-state Markov chains, we illustrate the approach on a sub-network designed from human glioma gene expression data and determine the joint steadystate probabilities for several groups of genes.

Transient phenomena for Markov chains and applications

1992 ◽  
Vol 24 (02) ◽  
pp. 322-342
Author(s):  
A. A. Borovkov ◽  
G. Fayolle ◽  
D. A. Korshunov
Keyword(s):  
Steady State ◽  
Weak Convergence ◽  
Markov Chains ◽  
Critical Behaviour ◽  
Transient Phenomena ◽  
Limit Distributions ◽  
Uniform Distributions

We consider a family of irreducible, ergodic and aperiodic Markov chains X(ε) = {X(ε) n, n ≧0} depending on a parameter ε > 0, so that the local drifts have a critical behaviour (in terms of Pakes' lemma). The purpose is to analyse the steady-state distributions of these chains (in the sense of weak convergence), when ε↓ 0. Under assumptions involving at most the existence of moments of order 2 + γ for the jumps, we show that, whenever X (0) is not ergodic, it is possible to characterize accurately these limit distributions. Connections with the gamma and uniform distributions are revealed. An application to the well-known ALOHA network is given.

Sign in / Sign up

Export Citation Format

Share Document