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Algebraic method of simplifying Boolean networks using semi‐tensor product of Matrices
Author(s) -
Yan Yongyi,
Yue Jumei,
Chen Zengqiang
Publication year - 2019
Publication title -
asian journal of control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.769
H-Index - 53
eISSN - 1934-6093
pISSN - 1561-8625
DOI - 10.1002/asjc.2125
Subject(s) - boolean network , logical matrix , and inverter graph , tensor product , boolean function , boolean algebra , product (mathematics) , algebraic number , matrix multiplication , attractor , mathematics , computer science , boolean data type , theoretical computer science , tensor (intrinsic definition) , product term , matrix (chemical analysis) , topology (electrical circuits) , boolean expression , algebra over a field , algorithm , two element boolean algebra , pure mathematics , combinatorics , algebra representation , mathematical analysis , chemistry , quantum , geometry , quantum mechanics , physics , organic chemistry , group (periodic table) , materials science , composite material
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.