TY - JOUR T1 - On Generating Optimal Sparse Probabilistic Boolean Networks with Maximum Entropy from a Positive Stationary Distribution 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), Entropy, Probabilistic Boolean Networks (PBNs), genetic regulatory networks, sparsity, stationary probability distribution, 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.