A Dynamical Boolean Network

We propose a Dynamical Boolean Network (DBN), which is a Boolean network whose set of states is fixed but whose transition matrix can change from one discrete time step to another. At any time step, the transition matrix of our DBN is of the formQ−1TQ, where T is a transition matrix defined in the course of the construction of our DBN and Q is the matrix representation of some permutation P of the states of that DBN. For each of several classes of such permutations, we carried out a number of simulations of a DBN. In one of the simulations, linearity appears to play a significant role in that, for all but two of the single-node rule vectors ~v associated with the transition matrix at some time step in that simulation, ~v is either a single-node rule vector of a Linear Virtual Boolean Network (LVBN) or the negation of such a rule vector. We also discuss the notions of a Probabilistic Boolean Network and a Hidden Markov Model—in both cases, in the context of using an arbitrary (though not necessarily one-to-one) function to label the states of a Virtual Boolean Network (VBN).