On Construction of Sparse Probabilistic Boolean Networks

2012 ◽  
Vol 2 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Xi Chen ◽  
Hao Jiang ◽  
Wai-Ki Ching
Keyword(s):  
Objective Function ◽  
Stationary Distribution ◽  
Transition Probability ◽  
Additional Term ◽  
Control Policy ◽  
Transition Probability Matrix ◽  
Boolean Networks ◽  
Entropy Rate ◽  
Sparse Solution ◽  
Probabilistic Boolean Networks

AbstractIn this paper we envisage building Probabilistic Boolean Networks (PBNs) from a prescribed stationary distribution. This is an inverse problem of huge size that can be subdivided into two parts — viz. (i) construction of a transition probability matrix from a given stationary distribution (Problem ST), and (ii) construction of a PBN from a given transition probability matrix (Problem TP). A generalized entropy approach has been proposed for Problem ST and a maximum entropy rate approach for Problem TP respectively. Here we propose to improve both methods, by considering a new objective function based on the entropy rate with an additional term of La-norm that can help in getting a sparse solution. A sparse solution is useful in identifying the major component Boolean networks (BNs) from the constructed PBN. These major BNs can simplify the identification of the network structure and the design of control policy, and neglecting non-major BNs does not change the dynamics of the constructed PBN to a large extent. Numerical experiments indicate that our new objective function is effective in finding a better sparse solution.

On Generating Optimal Sparse Probabilistic Boolean Networks with Maximum Entropy from a Positive Stationary Distribution

2012 ◽  
Vol 2 (4) ◽  
pp. 353-372 ◽  
Author(s):  
Hao Jiang ◽  
Xi Chen ◽  
Yushan Qiu ◽  
Wai-Ki Ching
Keyword(s):  
Maximum Entropy ◽  
Stationary Distribution ◽  
Sparse Matrix ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Boolean Networks ◽  
Efficiency And Effectiveness ◽  
Probabilistic Boolean Networks ◽  
Transition Probability Matrices ◽  
Probability Matrices

Abstract.To understand a genetic regulatory network, two popular mathematical models, Boolean Networks (BNs) and its extension Probabilistic Boolean Networks (PBNs) have been proposed. Here we address the problem of constructing a sparse Probabilistic Boolean Network (PBN) from a prescribed positive stationary distribution. A sparse matrix is more preferable, as it is easier to study and identify the major components and extract the crucial information hidden in a biological network. The captured network construction problem is both ill-posed and computationally challenging. We present a novel method to construct a sparse transition probability matrix from a given stationary distribution. A series of sparse transition probability matrices can be determined once the stationary distribution is given. By controlling the number of nonzero entries in each column of the transition probability matrix, a desirable sparse transition probability matrix in the sense of maximum entropy can be uniquely constructed as a linear combination of the selected sparse transition probability matrices (a set of sparse irreducible matrices). Numerical examples are given to demonstrate both the efficiency and effectiveness of the proposed method.

Construction of Probabilistic Boolean Networks from a Prescribed Transition Probability Matrix: A Maximum Entropy Rate Approach

2011 ◽  
Vol 1 (2) ◽  
pp. 132-154 ◽  
Author(s):  
Xi Chen ◽  
Wai-Ki Ching ◽  
Xiao-Shan Chen ◽  
Yang Cong ◽  
Nam-Kiu Tsing
Keyword(s):  
Inverse Problem ◽  
Maximum Entropy ◽  
Regulatory Networks ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
Genomic Research ◽  
Long Run ◽  
Probabilistic Boolean Networks

AbstractModeling genetic regulatory networks is an important problem in genomic research. Boolean Networks (BNs) and their extensions Probabilistic Boolean Networks (PBNs) have been proposed for modeling genetic regulatory interactions. In a PBN, its steady-state distribution gives very important information about the long-run behavior of the whole network. However, one is also interested in system synthesis which requires the construction of networks. The inverse problem is ill-posed and challenging, as there may be many networks or no network having the given properties, and the size of the problem is huge. The construction of PBNs from a given transition-probability matrix and a given set of BNs is an inverse problem of huge size. We propose a maximum entropy approach for the above problem. Newton's method in conjunction with the Conjugate Gradient (CG) method is then applied to solving the inverse problem. We investigate the convergence rate of the proposed method. Numerical examples are also given to demonstrate the effectiveness of our proposed method.

scholarly journals Markov Chains with Hybrid Repeating Rows - Upper-Hessenberg, Quasi-Toeplitz Structure of the Block Transition Probability Matrix

2008 ◽  
Vol 45 (01) ◽  
pp. 211-225 ◽  
Author(s):  
Alexander Dudin ◽  
Chesoong Kim ◽  
Valentina Klimenok
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Transition Probability ◽  
Sufficient Conditions ◽  
Transition Probability Matrix ◽  
Toeplitz Structure

In this paper we consider discrete-time multidimensional Markov chains having a block transition probability matrix which is the sum of a matrix with repeating block rows and a matrix of upper-Hessenberg, quasi-Toeplitz structure. We derive sufficient conditions for the existence of the stationary distribution, and outline two algorithms for calculating the stationary distribution.

scholarly journals Markov Chains with Hybrid Repeating Rows - Upper-Hessenberg, Quasi-Toeplitz Structure of the Block Transition Probability Matrix

2008 ◽  
Vol 45 (1) ◽  
pp. 211-225 ◽  
Author(s):  
Alexander Dudin ◽  
Chesoong Kim ◽  
Valentina Klimenok
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Transition Probability ◽  
Sufficient Conditions ◽  
Transition Probability Matrix ◽  
Toeplitz Structure

In this paper we consider discrete-time multidimensional Markov chains having a block transition probability matrix which is the sum of a matrix with repeating block rows and a matrix of upper-Hessenberg, quasi-Toeplitz structure. We derive sufficient conditions for the existence of the stationary distribution, and outline two algorithms for calculating the stationary distribution.

scholarly journals Generating probabilistic Boolean networks from a prescribed stationary distribution

2010 ◽  
Vol 180 (13) ◽  
pp. 2560-2570 ◽  
Author(s):  
Shu-Qin Zhang ◽  
Wai-Ki Ching ◽  
Xi Chen ◽  
Nam-Kiu Tsing
Keyword(s):  
Stationary Distribution ◽  
Boolean Networks ◽  
Probabilistic Boolean Networks

Optimal control policy for probabilistic Boolean networks with hard constraints

2009 ◽  
Vol 3 (2) ◽  
pp. 90-99 ◽  
Author(s):  
W.-K. Ching ◽  
A.S. Wong ◽  
T. Akutsu ◽  
N.-K. Tsing ◽  
S.-Q. Zhang ◽  
...  
Keyword(s):  
Optimal Control ◽  
Control Policy ◽  
Boolean Networks ◽  
Hard Constraints ◽  
Probabilistic Boolean Networks

scholarly journals On a multilevel controlled bulk queueing system MX/Gr,R/1

1992 ◽  
Vol 5 (3) ◽  
pp. 237-260 ◽  
Author(s):  
Lev Abolnikov ◽  
Jewgeni H. Dshalalow
Keyword(s):  
Stationary Distribution ◽  
Queue Length ◽  
Transition Probability ◽  
Queueing Systems ◽  
Transition Probability Matrix ◽  
Queueing Process ◽  
Poisson Input ◽  
Service Delay ◽  
Associated Functions ◽  
Semi Markov Process

The authors introduce and study a class of bulk queueing systems with a compound Poisson input modulated by a semi-Markov process, multilevel control service time and a queue length dependent service delay discipline. According to this discipline, the server immediately starts the next service act if the queue length is not less than r; in this case all available units, or R (capacity of the server) of them, whichever is less, are taken for service. Otherwise, the server delays the service act until the number of units in the queue reaches or exceeds level r.The authors establish a necessary and sufficient criterion for the ergodicity of the embedded queueing process in terms of generating functions of the entries of the corresponding transition probability matrix and of the roots of a certain associated functions in the unit disc of the complex plane. The stationary distribution of this process is found by means of the results of a preliminary analysis of some auxiliary random processes which arise in the “first passage problem” of the queueing process over level r. The stationary distribution of the queueing process with continuous time parameter is obtained by using semi-regenerative techniques. The results enable the authors to introduce and analyze some functionals of the input and output processes via ergodic theorems. A number of different examples (including an optimization problem) illustrate the general methods developed in the article.

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