Matrix representation of Kauffman networks

Kauffman networks are a class of Boolean networks where each node has the same number of incoming connections. Despite the simplicity of such networks, they exhibit very complex behaviors and have been shown to be an appropriate model for certain gene regulatory networks. Kauffman networks are typically represented by Boolean logics for which no efficient analytical tools are available. The logical representation of Kauffman networks makes it extremely difficult to analyze their dynamic behaviors. Based on a recently developed tool named “semi-tensor product” for matrices, we propose a novel matrix representation for Kauffman networks. This matrix representation is essentially a linear discrete dynamic system, making it possible to analyze the dynamic behaviors of Kauffman networks using existing tools in dynamic systems. As an example of the advantages of using this matrix representation, we show how the number and length of attractors can be calculated efficiently which is an impossible task for the original logical representation. Some general properties of Kauffman networks are also discussed based on their matrix representation.