Volume 2, Issue 4
On Generating Optimal Sparse Probabilistic Boolean Networks with Maximum Entropy from a Positive Stationary Distribution

Hao Jiang, Xi Chen, Yushan Qiu & Wai-Ki Ching

East Asian J. Appl. Math., 2 (2012), pp. 353-372.

Published online: 2018-02

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  • 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.

  • Keywords

Boolean Networks (BNs) Entropy Probabilistic Boolean Networks (PBNs) genetic regulatory networks sparsity stationary probability distribution transition probability matrix

  • AMS Subject Headings

65C20 92B05

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COPYRIGHT: © Global Science Press

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@Article{EAJAM-2-353, author = {Hao Jiang, Xi Chen, Yushan Qiu and Wai-Ki Ching}, title = {On Generating Optimal Sparse Probabilistic Boolean Networks with Maximum Entropy from a Positive Stationary Distribution}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {2}, number = {4}, pages = {353--372}, 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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.191012.221112a}, url = {http://global-sci.org/intro/article_detail/eajam/10882.html} }
TY - JOUR T1 - On Generating Optimal Sparse Probabilistic Boolean Networks with Maximum Entropy from a Positive Stationary Distribution AU - Hao Jiang, Xi Chen, Yushan Qiu & Wai-Ki Ching JO - East Asian Journal on Applied Mathematics VL - 4 SP - 353 EP - 372 PY - 2018 DA - 2018/02 SN - 2 DO - http://doi.org/10.4208/eajam.191012.221112a UR - https://global-sci.org/intro/article_detail/eajam/10882.html KW - Boolean Networks (BNs) KW - Entropy KW - Probabilistic Boolean Networks (PBNs) KW - genetic regulatory networks KW - sparsity KW - stationary probability distribution KW - transition probability matrix AB -

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.

Hao Jiang, Xi Chen, Yushan Qiu & Wai-Ki Ching. (1970). On Generating Optimal Sparse Probabilistic Boolean Networks with Maximum Entropy from a Positive Stationary Distribution. East Asian Journal on Applied Mathematics. 2 (4). 353-372. doi:10.4208/eajam.191012.221112a
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