A Dynamical Boolean Network

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.

💡 Research Summary

The paper introduces a novel class of discrete‑time dynamical systems called Dynamical Boolean Networks (DBNs). A DBN is built on the concept of a Virtual Boolean Network (VBN), which is a Boolean network (BN) in which every node receives the entire set of nodes as its input. Consequently, any finite BN can be represented as a VBN by suitably extending its transition rules. In a VBN the state space consists of all binary vectors of length |V| (|V| = number of nodes) and the dynamics are captured by a transition matrix T that has exactly one “1” in each row, reflecting the deterministic next‑state mapping.

The key innovation of a DBN is to allow the transition matrix to change at each time step. For step k the matrix is defined as

 Tₖ = Qₖ⁻¹ T Qₖ,

where Qₖ is the permutation matrix associated with a randomly chosen permutation Pₖ of the state set. Because Qₖ is orthogonal, Tₖ is a similarity transform of T; thus the underlying graph structure (e.g., eigenvalues) is preserved while the mapping from states to next states is shuffled. This mechanism creates a “dynamic” transition rule without altering the fixed state space, thereby breaking the usual attractor‑only behavior of ordinary BNs.

The authors further discuss labeling functions L: 2^{|V|} → Ω that map internal states to observable symbols. By allowing L to be non‑injective, they connect DBNs to Probabilistic Boolean Networks (PBNs) and Hidden Markov Models (HMMs): the stochastic selection of transition rules in a PBN can be viewed as a random choice of permutation, while the hidden state in an HMM corresponds to the internal VBN state that is only indirectly observed through L.

To explore the behavior of DBNs, the paper conducts extensive simulations on the smallest non‑trivial case: a two‑node network (four possible states). Four families of permutations are examined:

1. Type 1 – No permutation (fixed T).
2. Type 2 – Pₖ is either the identity, a single 2‑cycle, or a product of two disjoint 2‑cycles.
3. Type 3 – Pₖ is drawn uniformly from all 4! = 24 permutations of the four states.
4. Type 4 – Pₖ is generated from a labeling function that encodes additional structure.

Each simulation consists of 1,000 independent trials, each trial running for 10,000 discrete steps. The authors record which “rule vectors” (the concatenation of the two single‑node transition tables) are visited. For a two‑node DBN there are 2⁸ = 256 possible rule vectors.

Results show that in Types 1 and 2 many rule vectors (81 of them) are never visited after the fifth step, indicating that limited permutation classes severely restrict the reachable dynamics. In contrast, Type 3 visits all 256 vectors in every trial, and Type 4 visits all vectors in slightly more than half of the trials. Notably, in Type 4 the six most frequently visited rule vectors correspond to six single‑node transition rules: three are the linear rules of a Linear VBN (LVBN) (i.e., XOR‑type updates) and the other three are their logical negations. This suggests that linear update functions are naturally favored when the transition matrix is repeatedly conjugated by permutations derived from a labeling scheme.

The authors interpret this as evidence that the similarity transformation preserves linear structure, making linear rules more “stable” under dynamic permutation. They also point out that the labeling‑based permutation introduces a feedback loop between the observable layer and the underlying VBN, reminiscent of the perpetual loop described in Internal Measurement theory.

While the experimental work is limited to two nodes, the paper highlights several important implications:

• Conceptual novelty – DBNs provide a framework where the deterministic transition rule itself evolves, bridging deterministic BNs, stochastic PBNs, and hidden‑state HMMs.
• Role of linearity – The prevalence of linear rules under certain permutation schemes may inform the design of robust synthetic gene circuits or fault‑tolerant digital systems.
• Scalability concerns – The number of possible permutations grows factorially with the number of states, making exhaustive exploration infeasible for larger networks. Future work must devise structured permutation families (e.g., group‑theoretic subgroups) or approximation methods.
• Potential applications – Modeling biological regulatory networks where the wiring can rewire over time, adaptive control systems, and cryptographic constructions where the transition function changes dynamically.

In conclusion, the paper establishes DBNs as a flexible extension of Boolean network theory, demonstrates through simulation that dynamic permutation of transition matrices can lead to rich, non‑attractor dynamics, and uncovers a surprising bias toward linear update rules when permutations are linked to a labeling function. Further research is needed to scale the approach, analyze its theoretical properties (e.g., ergodicity, mixing times), and apply it to real‑world complex systems.