Logical Matrix Factorization With Application to Topological Structure Analysis of Boolean Network

This note investigates the logical matrix factorization with application to the topological structure analysis of Boolean networks. First, the concepts of both factorization and rank are defined for logical matrices, and two factorization problems are then studied. Using nonsingular logical matrix transformations, several necessary and sufficient conditions for the factorization of a given logical matrix are presented. Second, the logical matrix factorization is applied to the topological structure analysis of a given Boolean network, and a size-reduced structure-equivalent logical network is constructed for the given system. It is shown that the topological structure (including all the fixed points and cycles) of the resulting size-reduced logical network is the same as that of the original Boolean network. The study of an illustrative example shows that the new results presented in this note are effective in analyzing the topological structure of Boolean networks.