[Paper Review] Circuits, Attractors and Reachability in Mixed-K Kauffman Networks

Ken A. Hawick, H. A. James+1 more|ArXiv.org|Nov 15, 2007

Gene Regulatory Network Analysis34 references3 citations

TL;DR

This paper introduces a mixed-K Kauffman network model to study structural circuits and their relationship to attractor dynamics, revealing a previously unreported phase transition at K≈1.5 that influences circuit growth and reachability. Using fast circuit enumeration, the authors establish that circuit count serves as a lower bound for attractor number, linking network structure directly to dynamical complexity in random Boolean networks.

ABSTRACT

The growth in number and nature of dynamical attractors in Kauffman NK network models are still not well understood properties of these important random boolean networks. Structural circuits in the underpinning graph give insights into the number and length distribution of attractors in the NK model. We use a fast direct circuit enumeration algorithm to study the NK model and determine the growth behaviour of structural circuits. This leads to an explanation and lower bound on the growth properties and the number of attractor loops and a possible K-relationship for circuit number growth with network size N. We also introduce a mixed-K model that allows us to explore between pairs of integer K values in Kauffman-like systems. We find that the circuits' behaviour is a useful metric in identifying phase transitional behaviour around the critical connectivity in that model too. We identify an intermediate phase transition in circuit growth behaviour at K_S approximately 1.5, that is distinct from both the percolation transition at K_P = 1 and the Kauffman transition at K_C = 2. We relate this transition to mutual node reachability within the giant component of nodes.

Motivation & Objective

• To understand the poorly understood growth of dynamical attractors in Kauffman NK networks.
• To investigate the structural role of circuits in determining attractor number and length distribution.
• To introduce a mixed-K model that allows continuous variation of mean connectivity ⟨K⟩ between integer K values.
• To identify and characterize new phase transitions in network connectivity beyond the known K=1 (percolation) and K=2 (chaotic) thresholds.
• To examine how mutual node reachability and circuit count co-evolve with increasing connectivity.

Proposed method

• Implementing a fast direct circuit enumeration algorithm to compute structural circuits in directed random graphs.
• Defining a mixed-K model where each node's in-degree is drawn from a discrete distribution centered on a non-integer ⟨K⟩, enabling continuous exploration of connectivity.
• Using synchronous updating of boolean functions on nodes, with truth tables assigned uniformly at random.
• Measuring reachability via all-pairs shortest path distances, with unreachable pairs contributing zero to the average to reflect structural limitations.
• Analyzing the growth of circuits as a function of network size N and mean connectivity ⟨K⟩, focusing on the regime 1 < ⟨K⟩ < 2.
• Comparing circuit counts to attractor dynamics, positing circuits as a lower bound on attractor number.

Experimental results

Research questions

• RQ1 How does the number of structural circuits in a random directed graph scale with network size N and mean connectivity ⟨K⟩?
• RQ2 Does a phase transition in circuit growth occur at a connectivity value distinct from K=1 (percolation) and K=2 (chaotic transition)?
• RQ3 What is the relationship between mutual node reachability and circuit count in the intermediate connectivity regime?
• RQ4 Can circuit enumeration serve as a lower bound for the number of attractors in random Boolean networks?
• RQ5 How does the mixed-K model reveal structural transitions not visible in standard integer-K NK models?

Key findings

• A structural transition in circuit growth is identified at K≈1.5, distinct from the percolation transition at K=1 and the chaotic transition at K=2.
• The number of structural circuits grows approximately as an exponential function of the number of arcs NA ≈ K·N, with the exponent dependent on ⟨K⟩.
• At K≈1.5, the network achieves maximal traversal distance across pairs of nodes, indicating a peak in effective connectivity for reachability.
• Mutual reachability among nodes remains incomplete even at high K, with persistent 'islands of directed disconnection' in the network.
• The fraction of nodes in the giant component sharpens with increasing network size, indicating a more defined percolation transition.
• Circuit count provides a robust lower bound on the number of attractors in both standard and mixed-K Kauffman networks, suggesting a fundamental structural origin of dynamical complexity.