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Analysis of random Boolean networks using the average sensitivity
, 2008
"... In this work we consider random Boolean networks that provide a general model for genetic regulatory networks. We extend the analysis of James Lynch who was able to proof Kauffman’s conjecture that in the ordered phase of random networks, the number of ineffective and freezing gates is large, where ..."
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In this work we consider random Boolean networks that provide a general model for genetic regulatory networks. We extend the analysis of James Lynch who was able to proof Kauffman’s conjecture that in the ordered phase of random networks, the number of ineffective and freezing gates is large, where as in the disordered phase their number is small. Lynch proved the conjecture only for networks with connectivity two and nonuniform probabilities for the Boolean functions. We show how to apply the proof to networks with arbitrary connectivity K and to random networks with biased Boolean functions. It turns out that in these cases Lynch’s parameter λ is equivalent to the expectation of average sensitivity of the Boolean functions used to construct the network. Hence we can apply a known theorem for the expectation of the average sensitivity. In order to prove the results for networks with biased functions, we deduct the expectation of the average sensitivity when only functions with specific connectivity and specific bias are chosen at random.
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.
Are selforganizing biochemical networks emergent?
"... Abstract. Biochemical networks are often called upon to illustrate emergent properties of living systems. In this contribution, I question such emergentist claims by means of theoretical work on genetic regulatory models and random Boolean networks. If the existence of a critical connectivity Kc of ..."
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Abstract. Biochemical networks are often called upon to illustrate emergent properties of living systems. In this contribution, I question such emergentist claims by means of theoretical work on genetic regulatory models and random Boolean networks. If the existence of a critical connectivity Kc of such networks has often been coined 'emergent' or 'irreducible', I propose on the contrary that the existence of a critical connectivity Kc is indeed mathematically explainable in network theory. This conclusion also applies to many other types of formal networks and weakens the emergentist claim attached to biomolecular networks, and by extension to living systems.
STABILITY IN RANDOM BOOLEAN CELLULAR AUTOMATA ON THE INTEGER LATTICE
, 2007
"... We consider random boolean cellular automata on the integer lattice, i.e., the cells are identified with the integers from 1 to N. The behaviour of the automaton is mainly determined by the support of the random variable that selects one of the sixteen possible Boolean rules, independently for each ..."
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We consider random boolean cellular automata on the integer lattice, i.e., the cells are identified with the integers from 1 to N. The behaviour of the automaton is mainly determined by the support of the random variable that selects one of the sixteen possible Boolean rules, independently for each cell. A cell is said to stabilize if it will not change its state anymore after some time. We classify the random boolean automata according to the positivity of their probability of stabilization. Here is an example of a consequence of our results: if the support contains at least 5 rules, then asymptotically as N tends to infinity the probability of stabilization is positive, whereas there exist random boolean cellular automata with 4 rules in their support for which this probability tends to 0.
Commentary on “Emergence in Biomolecular Networks?”
"... We examine several versions of the notion of “emergence ” in the context of random Boolean networks. Certain properties of the behavior of these networks have often been claimed to be emergent. We discuss the strengths of these claims under different interpretations of emergence and different stand ..."
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We examine several versions of the notion of “emergence ” in the context of random Boolean networks. Certain properties of the behavior of these networks have often been claimed to be emergent. We discuss the strengths of these claims under different interpretations of emergence and different standards of mathematical proof. 1