#### Computational Science Technical Note CSTN-046

#
Circuits, Attractors and Reachability in Mixed-K Kauffman Networks

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K. A. Hawick, C. J. Scogings and H. A. James

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Archived November 2007

**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.

**Keywords:**
PACS 87.15Aa Theory and modelling; 89.75Hc Networks and geneological trees.

**Full Document Text:**
PDF version.

**Citation Information:** BiBTeX database for CSTN Notes.

**BiBTeX reference:**

@TECHREPORT{CSTN-046,
author = {K. A. Hawick and H. A. James and C. J. Scogings},
title = {{Circuits, Attractors and Reachability in Mixed-K Kauffman Networks}},
institution = {Massey University},
year = {2007},
number = {CSTN-046; arXiv:0711.2426},
month = {15 November},
timestamp = {2007.11.12}
}

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