East Asian Journal on Applied Mathematics, 2 (2012), pp. 1-18. On Construction of Sparse Probabilistic Boolean Networks Xi Chen 1*, Hao Jiang 1, Wai-Ki Ching 11 Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, Received 3 May 2011; Accepted (in revised version) 6 September 2011 Available online 10 February 2012 doi:10.4208/eajam.030511.060911a Abstract In 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 $L_{\alpha}$-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. AMS subject classifications: 65C20, 92B05 Key words: Probabilistic Boolean Networks, entropy, stationary distribution, sparsity, transition probability matrix. *Corresponding author. Email: dlkcissy@hku.hk (Xi Chen),