[Paper Review] Phase Transitions in Random Boolean Networks with Different Updating Schemes

Carlos Gershenson|ArXiv.org|Nov 5, 2003

Gene Regulatory Network AnalysisBiochemistry, Genetics and Molecular Biology14 references21 citations

TL;DR

This study investigates phase transitions in Random Boolean Networks (RBNs) under various updating schemes—synchronous, asynchronous, deterministic, and non-deterministic—using computer simulations to measure sensitivity to initial conditions. The key finding is that all RBN types exhibit a critical transition from order to chaos at an average connectivity of 1 < k < 3, with network size having a stronger influence on the critical value than the updating scheme.

ABSTRACT

In this paper we study the phase transitions of different types of Random Boolean networks. These differ in their updating scheme: synchronous, semi-synchronous, or asynchronous, and deterministic or non-deterministic. It has been shown that the statistical properties of Random Boolean networks change considerable according to the updating scheme. We study with computer simulations sensitivity to initial conditions as a measure of order/chaos. We find that independently of their updating scheme, all network types have very similar phase transitions, namely when the average number of connections of nodes is between one and three. This critical value depends more on the size of the network than on the updating scheme.

Motivation & Objective

• To determine whether different updating schemes (synchronous, asynchronous, deterministic, non-deterministic) alter the dynamical phase (ordered, critical, chaotic) of Random Boolean Networks.
• To investigate how the statistical properties of RBNs—such as attractor number and basin size—vary across updating schemes.
• To assess whether the critical connectivity threshold (k) for phase transitions depends on the updating scheme or network size.
• To explore the implications of these findings for modeling complex systems, particularly in biological and cognitive contexts.
• To evaluate the functional advantages of contextual updating (e.g., DGARBNs) in enabling information processing without shifting into chaos.

Proposed method

• Simulated Random Boolean Networks (RBNs) with fixed topology and logic functions, varying only the updating scheme.
• Used sensitivity to initial conditions as a quantitative measure of order versus chaos, tracking divergence of trajectories from slightly different initial states.
• Evaluated four RBN types: CRBN (synchronous deterministic), ARBN (asynchronous non-deterministic), DARBN (asynchronous deterministic), and GARBN/DGARBN (semi-synchronous, with deterministic or non-deterministic node selection).
• Varying the average connectivity (k) from 0 to 5 across network sizes (n = 200, 500, 1000) to identify phase transition points.
• Applied statistical analysis to measure average divergence rates and identify critical k values where behavior shifts from ordered to chaotic.
• Compared results across updating schemes and network sizes to isolate the influence of each factor on phase transitions.

Experimental results

Research questions

• RQ1 Does the updating scheme (synchronous, asynchronous, deterministic, non-deterministic) significantly alter the phase transition point in Random Boolean Networks?
• RQ2 How does network size influence the critical connectivity (k) at which RBNs transition from ordered to chaotic dynamics?
• RQ3 To what extent do statistical properties like attractor count and basin size differ across updating schemes, despite similar phase transitions?
• RQ4 Can deterministic asynchronous RBNs (DARBNs) and generalized asynchronous RBNs (GARBNs) achieve similar dynamical behavior to synchronous RBNs (CRBNs) in terms of phase transitions?
• RQ5 What functional advantages arise from using context-dependent updating (e.g., DGARBNs) in processing information without entering the chaotic regime?

Key findings

• All RBN types—regardless of updating scheme—exhibit a phase transition from ordered to chaotic dynamics when the average connectivity k is between 1 and 3.
• The precise critical value of k depends more on network size than on the updating scheme, with larger networks showing transitions closer to k = 1 and smaller ones closer to k = 3.
• For n = 200, networks with k ≤ 1 are typically static (ordered), while those with k ≥ 2 are predominantly chaotic, regardless of updating scheme.
• Asynchronous and deterministic updating schemes produce very similar sensitivity to initial conditions, indicating that synchronicity is more influential than determinism in phase transitions.
• Despite differences in attractor statistics (e.g., number and length), point attractors remain identical across all RBN types, as they are independent of update timing.
• DGARBNs can process more information via complex updating periods without shifting into chaos, suggesting an evolutionary advantage in contextual information use.