Boolean Networks: Coding, Linearizing and Dynamics

In this paper, an effective scheme is proposed for coding $n$-node Boolean networks. The scheme can uniquely designate a distinguished integer in the range from $0$ to $(2^{n\times2^n}-1)$ for any a given Boolean network. At the same time, a linearized matrix is obtained for any a given Boolean network. The linearized matrix depends only on the information hidden in the logical table of the given network. By analyzing the linearized matrix corresponding to the given network, we can easily deal with the dynamics of the network such as the number of the fixed points and the numbers of all possible circles of different lengths, basins of attraction of all attractors, and so on.