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Structural Circuits and Attractors in Kauffman Networks
"... There has been some ambiguity about the growth of attractors in Kauffman networks with network size. Some recent work has linked this to the role and growth of circuits or loops of boolean variables. Using numerical methods we have investigated the growth of structural circuits in Kauffman networks ..."
Abstract

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There has been some ambiguity about the growth of attractors in Kauffman networks with network size. Some recent work has linked this to the role and growth of circuits or loops of boolean variables. Using numerical methods we have investigated the growth of structural circuits in Kauffman networks and suggest that the exponential growth in the number of structural circuits places a lower bound on the complexity of the growth of boolean dependency loops and hence of the number of attractors. We use a fast and exact circuit enumeration method that does not rely on sampling trajectories. We also explore the role of structural selfedges, or selfinputs in the NKmodel, and how they affect the number of structural circuits and hence of attractors.
Circuits, Attractors and Reachability in MixedK Kauffman Networks
, 2007
"... 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 ..."
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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 Krelationship for circuit number growth with network size N. We also introduce a mixedK model that allows us to explore N 〈K 〉 between pairs of integer K values in Kauffmanlike 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 = KS ≈ 1.5, that is distinct from both the percolation transition at KP ≡ 1 and the Kauffman transition at KC ≡ 2. We relate this transition to mutual node reachability within the giant component of nodes.