[Paper Review] Contextual Random Boolean Networks

Carlos Gershenson, Jan Broekaert+1 more|ArXiv.org|Mar 10, 2003

Gene Regulatory Network Analysis16 references4 citations

TL;DR

This paper introduces Contextual Random Boolean Networks (CRBNs) using deterministic generalized asynchronous updating, where context—defined by node update periods—drastically alters network dynamics. By modeling uncertainty in context as a statistical mixture of pure contexts, the network exhibits non-Kolmogorovian, quantum-like probabilistic behavior, transforming states into potentiality states that collapse to attractors with context-dependent probabilities.

ABSTRACT

We propose the use of Deterministic Generalized Asynchronous Random Boolean Networks [Gershenson, 2002] as models of contextual deterministic discrete dynamical systems. We show that changes in the context have drastic effects on the global properties of the same networks, namely the average number of attractors and the average percentage of states in attractors. We introduce the situation where we lack knowledge on the context as a more realistic model for contextual dynamical systems. We notice that this makes the network non-deterministic in a specific way, namely introducing a non-Kolmogorovian quantum-like structure for the modelling of the network [Aerts, 1986]. In this case, for example, a state of the network has the potentiality (probability) of collapsing into different attractors, depending on the specific form of lack of knowledge on the context.

Motivation & Objective

• To model contextual deterministic discrete dynamical systems using deterministic generalized asynchronous RBNs.
• To investigate how changes in the context (node update periods) affect global network properties like attractor count and attractor basin size.
• To introduce a realistic model of contextuality by representing lack of knowledge about context as a statistical mixture of pure contexts.
• To demonstrate that such mixed contexts induce non-Kolmogorovian, quantum-like probability structures in network dynamics.
• To explore the implications of this framework for modeling biological and cognitive systems with context-dependent behavior.

Proposed method

• Define pure contexts as deterministic update schedules (periods) for each node in a Random Boolean Network (RBN).
• Model real-world context uncertainty by forming mixed contexts as probability distributions over a set of pure contexts.
• Simulate network dynamics under each pure context to determine attractor structure and basin sizes.
• Analyze the resulting probabilistic behavior under mixed contexts, where a state has potentiality to collapse into different attractors.
• Use the mathematical framework of non-Kolmogorovian probability to characterize the resulting dynamics.
• Apply the model to small networks to illustrate the emergence of quantum-like interference and contextuality in state evolution.

Experimental results

Research questions

• RQ1 How do changes in the update period context affect the average number of attractors and the percentage of states in attractors in RBNs?
• RQ2 What happens to network dynamics when context is uncertain, modeled as a statistical mixture of pure contexts?
• RQ3 Can the resulting probabilistic structure of state evolution under mixed contexts be described as non-Kolmogorovian, akin to quantum mechanics?
• RQ4 How does the introduction of mixed contexts transform deterministic states into potentiality states with context-dependent collapse probabilities?
• RQ5 Does the network exhibit phase transitions between order, chaos, and complexity under varying contextual influences?

Key findings

• Changing the context (update period schedule) drastically alters the global properties of the RBN, including the average number of attractors and the percentage of states in attractors.
• Networks under pure contexts (deterministic updating) show that the percentage of states in attractors diminishes exponentially with network size, regardless of connectivity or update scheme.
• Mixed contexts—statistical mixtures of pure contexts—introduce non-deterministic dynamics where states become potentiality states, collapsing to different attractors with probabilities determined by the weights of the pure contexts.
• The probability structure induced by mixed contexts is non-Kolmogorovian, exhibiting quantum-like features such as contextuality and interference, even though the underlying dynamics remain deterministic.
• The model reveals that restricted temporal contexts (small maximum update periods) resemble synchronous RBNs, while unrestricted contexts resemble non-deterministic asynchronous RBNs in behavior.
• The framework suggests the possibility of chaotic, complex, and ordered regimes under non-deterministic contextuality, with potential for phase transitions dependent on context, despite the non-linearity of the dynamics.