Highly-sensitive measure of complexity captures boolean networks regimes and temporal order more optimally

TL;DR

In this work, several random Boolean networks are generated and analyzed from two characteristics: their time evolution diagram and their transition diagram, which are useful for the introduction of both Algorithmic Complexity and Perturbation Analysis in the context of Boolean networks, and their potential applications in regulatory network models.

Abstract

In this work, several random Boolean networks (RBN) are generated and analyzed from two characteristics: their time evolution diagram and their transition diagram. For this purpose, its randomness is estimated using three measures, of which Algorithmic Complexity is capable of both a) revealing transitions towards the chaotic regime in a more marked way, and b) disclosing the algorithmic contribution of certain states to the transition diagram and their relationship with the order they occupy in the temporal evolution of the respective RBN. The results obtained from both types of analysis are useful for the introduction of both Algorithmic Complexity and Perturbation Analysis in the context of Boolean networks, and their potential applications in regulatory network models.

Figures (7)

• Figure 1: A specific Boolean network. Left: connectivity graph with $N=4$ and $k=2$. Medium: connectivity matrix and truth table. Right: transition diagram between all $2^{4}=16$ possible states, constructed from the connectivity matrix and the truth table; steady states or attractors are depicted in black.
• Figure 2: Time evolution diagrams (top) and their respective state transition graphs (bottom) for some RBNs with $N=500$ nodes and $k=5$. The RBNs share the same initial state and the same connectivity matrix, both generated randomly from uniform probability distributions.
• Figure 3: Top: randomness estimates of RBNs with $N=500$ and $k=5$ as a function of the parameter $p$. Bottom: estimates of average algorithmic complexity from samples of size 10. Vertical dotted lines represent the theoretical critical values of $p = 0.113$ and $p = 0.887$ at the edge of chaos. All RBNs share the same initial state and the same connectivity matrix, both generated randomly from uniform probability distributions.
• Figure 4: Left: Plots of boolean functions' BDM vs $p$. Right: Plots of time evolution diagrams' BDM vs $p$. Vertical dotted lines mark the critical $p$ values in each series. All RBNs share the same initial state, and all RBNs for a given in-degree $k$ share the same connectivity matrix; both types of objects were generated randomly from uniform probability distributions.
• Figure 5: Randomness estimates of RBNs with $N=500$ and $k=5$ as a function of the parameter $p$. Vertical dotted lines represent the theoretical critical values of $p = 0.113$ and $p = 0.887$ at the edge of chaos. All RBNs share the same initial state and the same connectivity matrix. The connectivity matrix was generated randomly from a binomial probability distribution with $p=0.5$ and $n=499$.