Algebraic method of simplifying Boolean networks using semi‐tensor product of Matrices

Abstract

Abstract The huge state space of large Boolean networks makes analysis and synthesis difficult. This paper, using a new matrix analysis tool called semi‐tensor product of matrices, to explain a simplification method of Boolean networks in a mathematical manner. The idea consists of two steps. First, remove the nodes whose logical dynamics are independent of themselves directly; second, use the logical functions (LFs) of the removed nodes to substitute for their corresponding variables in the LFs of other nodes; such nodes evolve directly with both themselves and the removed nodes. We discover that the simplified and original Boolean networks share some important topological structures such as attractor cycles, steady states and paths. An algebraic algorithm is provided to find all of the cycles and steady states of simplified Boolean networks. Finally we apply the results to the metastatic melanoma network to check the effect of the simplification method.