A Dynamical Boolean Network

TL;DR

This paper introduces a Dynamical Boolean Network (DBN) with a fixed state set and adaptable transition matrices, exploring its properties through simulations and discussing related probabilistic models.

Contribution

The paper proposes a novel DBN model with transition matrices that change via permutations, and demonstrates its behavior through extensive simulations of two-node networks.

Findings

Linear rules are prominent in the transition rules observed.

Simulations show the influence of linearity in the network dynamics.

The model connects to Probabilistic Boolean Networks and Hidden Markov Models.

Abstract

We propose a Dynamical Boolean Network (DBN), which is a Virtual Boolean Network (VBN) whose set of states is fixed but whose transition matrix can change from one discrete time step to another. The transition matrix $T_{k}$ of our DBN for time step $k$ is of the form $Q^{-1}TQ$, where $T$ is a transition matrix (of a VBN) defined at time step $k$ in the course of the construction of our DBN and $Q$ is the matrix representation of some randomly chosen permutation $P$ of the states of our DBN. For each of several classes of such permutations, we carried out a number of simulations of a DBN with two nodes; each of our simulations consisted of 1,000 trials of 10,000 time steps each. In one of our simulations, only six of the 16 possible single-node transition rules for a VBN with two nodes were visited a total of 300,000 times (over all 1,000 trials). In that simulation, linearity appears to play a significant role in that three of those six single-node transition rules are transition rules of a Linear Virtual Boolean Network (LVBN); the other three are the negations of the first three. 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 VBN.