Dynamics of critical Kauffman networks under asynchronous stochastic update

Area of Science:

• Complex systems
• Theoretical computer science
• Network dynamics

Background:

• Boolean networks are models for gene regulatory networks and other complex systems.
• Understanding the dynamics of these networks, particularly their attractors, is crucial for predicting system behavior.
• Previous studies often focused on synchronous updates, yielding different results.

Purpose of the Study:

• To investigate the behavior of attractors in critical Boolean networks under asynchronous stochastic updates.
• To compare these dynamics with those observed under synchronous updates.
• To characterize the scaling of attractor number and size with system size.

Main Methods:

• Analysis of critical Boolean networks with asynchronous stochastic updates.
• Mathematical modeling to derive scaling laws for attractor properties.
• Comparison of results with theoretical predictions for synchronous updates.

Main Results:

• The mean number of attractors grows as a power law with system size under asynchronous updates.
• The mean size of attractors increases as a stretched exponential with system size.
• This contrasts sharply with synchronous updates, where attractor number grows faster than any power law.

Conclusions:

• Asynchronous updates in critical Boolean networks lead to distinct scaling behaviors for attractors compared to synchronous updates.
• The power-law and stretched exponential scaling provide insights into the stability and complexity of these networks.
• These findings have implications for understanding biological systems and designing robust computational networks.