[Paper Review] Phase Transition in Random Networks with Multiple States

Ricard V. Solé, Bartolo Luque+1 more|arXiv (Cornell University)|Jul 23, 1999

Gene Regulatory Network Analysis2 references13 citations

TL;DR

This paper investigates phase transitions in random networks with S discrete states, generalizing random Boolean networks to model systems like genetic regulatory networks or multi-state agent collectives. Using a mean-field approach, it derives critical connectivity thresholds separating ordered and chaotic regimes, identifying a sharp transition at a specific S-dependent critical point, with implications for biological and complex systems stability.

ABSTRACT

The critical boundaries separating ordered from chaotic behavior in randomly wired S-state networks are calculated. These networks are a natural generalization of random Boolean nets and are proposed as on extended approach to genetic regulatory systems, sets of cells in different states or collectives of agents engaged into a set of S possible tasks. A order parameter for the transition is computed and analysed. The relevance of these networks to biology, their relationships with standard cellular automata and possible extensions are outlined.

Motivation & Objective

• To generalize random Boolean networks to S-state networks for modeling multi-state biological and social systems.
• To identify critical connectivity thresholds that separate ordered from chaotic dynamics in such networks.
• To develop a theoretical framework for analyzing stability and phase transitions in complex, randomly wired systems.
• To explore the relevance of these networks to real-world systems like gene regulatory networks and collective agent behavior.
• To extend the understanding of dynamical regimes in complex systems beyond binary-state models.

Proposed method

• Adopting a mean-field approximation to analyze the dynamics of randomly wired S-state networks.
• Defining a transition parameter based on the average connectivity and S-state transition probabilities.
• Deriving analytical expressions for the critical connectivity k_c that separates ordered and chaotic phases.
• Using a stability analysis of the fixed points to determine the order parameter of the transition.
• Extending the framework of Kauffman's random Boolean networks to S states using generalized transition rules.
• Computing the Lyapunov exponent to quantify dynamical stability and identify the phase boundary.

Experimental results

Research questions

• RQ1 What are the critical connectivity thresholds that separate ordered from chaotic behavior in S-state random networks?
• RQ2 How does the number of states S affect the location and nature of the phase transition?
• RQ3 What is the order parameter that characterizes the transition between ordered and chaotic regimes?
• RQ4 How do the dynamics of S-state networks compare to those of standard random Boolean networks?
• RQ5 What are the implications of this phase transition for the stability of biological regulatory systems?

Key findings

• The critical connectivity k_c scales inversely with the number of states S, indicating that higher S values stabilize the network against chaos.
• A sharp phase transition is observed at a well-defined k_c, separating ordered and chaotic dynamical regimes.
• The order parameter, derived from the stability of fixed points, confirms a continuous transition with a clear critical threshold.
• The Lyapunov exponent crosses zero at k_c, confirming the transition from stable to unstable dynamics.
• The model generalizes Kauffman's random Boolean network framework to S states, enabling broader biological and sociological applications.
• The analytical framework successfully predicts the onset of chaotic behavior in complex, randomly wired systems.