Feedback topology and XOR-dynamics in Boolean networks with varying input structure

Area of Science:

• Complex systems
• Network science
• Theoretical computer science

Background:

• Random Boolean networks (RBNs) are models for complex systems.
• Understanding the relationship between network topology and dynamics is crucial.
• Kauffman networks highlight the role of feedback in RBN dynamics.

Purpose of the Study:

• Analyze the dynamics of fixed in-degree random Boolean networks.
• Investigate the impact of the input-receiving node fraction (gamma) on network behavior.
• Characterize the phase transition in these networks.

Main Methods:

• Analytical and numerical investigation of network dynamics.
• Utilized a parallel XOR updating scheme for tractability and richness.
• Employed graph decimation algorithms (Leaf Removal) to identify feedback regions.

Main Results:

• Derived analytical formulas for dynamics based on topological feedback structure.
• Identified a phase transition controlled by gamma, separating treelike from feedback-rich regions.
• Observed that networks near the transition have feedback components composed of disjoint loops.

Conclusions:

• Network dynamics in XOR-updating RBNs are strongly dictated by topology.
• The parameter gamma controls a transition from treelike structures to emergent feedback.
• Topological analysis of feedback loops allows for estimation of maximum network periods.