The phase diagram of random threshold networks

Szejka, Agnes; Mihaljev, Tamara; Drossel, Barbara

doi:10.1088/1367-2630/10/6/063009

Cited by 36 publications

(57 citation statements)

References 13 publications

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“…Such 2-input threshold functions have been studied by Greil and Drossel [12] and Szejka et al [13] and are classified as biologically meaningful by Raeymaekers [14]. We now use Theorem 1 to show that random threshold functions lead to criticality for any indegree distribution.…”

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confidence: 90%

Influence and Dynamic Behavior in Random Boolean Networks

et al. 2011

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We present a rigorous mathematical framework for analyzing dynamics of a broad class of Boolean network models. We use this framework to provide the first formal proof of many of the standard critical transition results in Boolean network analysis, and offer analogous characterizations for novel classes of random Boolean networks. We precisely connect the short-run dynamic behavior of a Boolean network to the average influence of the transfer functions. We show that some of the assumptions traditionally made in the more common mean-field analysis of Boolean networks do not hold in general. For example, we offer some evidence that imbalance, or expected internal inhomogeneity, of transfer functions is a crucial feature that tends to drive quiescent behavior far more strongly than previously observed.Introduction. Complex systems can usually be represented as a network of interdependent functional units. Boolean networks were proposed by Kauffman as models of genetic regulatory networks [1, 2] and have received considerable attention across several scientific disciplines. They model a variety of complex phenomena, particularly in theoretical biology and physics [3][4][5][6][7][8].A Boolean network N with n nodes can be described by a directed graph G = (V, E) and a set of transfer functions. We use V and E to denote the sets of nodes and edges respectively, and denote the indegree of node i by K i . Each node i is assigned a K i -ary Boolean function f i : {−1, +1} K i → {−1, +1}, termed transfer function. If the state of node i at time t is x i (t), its state at time t + 1 is described by

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confidence: 90%

Influence and Dynamic Behavior in Random Boolean Networks

et al. 2011

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“…Indeed, one may use the so-called annealed approximation, which is a sort of mean field approximation-that holds for annealed networks [11] with an infinite number of nodes-to estimate b and hence the value of SA. This approach can give reasonable guesses also for quenched systems [11,39]. Nevertheless, it takes averages and therefore ignores the possible peculiar behaviour on individual attractors; moreover, it cannot take into account the effect of the finite number of nodes that may be often non-negligible.…”

Section: Linking Static and Dynamical Measuresmentioning

confidence: 99%

Dynamical regimes in non-ergodic random Boolean networks

et al. 2016

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Random boolean networks are a model of genetic regulatory networks that has proven able to describe experimental data in biology. Random boolean networks not only reproduce important phenomena in cell dynamics, but they are also extremely interesting from a theoretical viewpoint, since it is possible to tune their asymptotic behaviour from order to disorder. The usual approach characterizes network families as a whole, either by means of static or dynamic measures. We show here that a more detailed study, based on the properties of system’s attractors, can provide information that makes it possible to predict with higher precision important properties, such as system’s response to gene knock-out. A new set of principled measures is introduced, that explains some puzzling behaviours of these networks. These results are not limited to random Boolean network models, but they are general and hold for any discrete model exhibiting similar dynamical characteristics

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“…If one furthermore assumes that the activatory and inhibitory regulators of a gene interact in an additive fashion, one can model genetic networks by so-called random threshold networks (Rohlf and Bornholdt, 2002;Szejka et al, 2008). Here, the update function of a node is a weighted sum over its inputs (with positive and negative weights depending on whether the input is an activator or inhibitor), which is thresholded to yield a binary output.…”

Section: Biologically Meaningful Update Functionsmentioning

confidence: 99%

Biologically meaningful update rules increase the critical connectivity of generalized Kauffman networks

Wittmann¹,

Marr²,

Theis³

2010

Journal of Theoretical Biology

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We generalize random Boolean networks by softening the hard binary discretization into multiple discrete states. These multistate networks are generic models of gene regulatory networks, where each gene is known to assume a finite number of functionally different expression levels. We analytically determine the critical connectivity that separates the biologically unfavorable frozen and chaotic regimes. This connectivity is inversely proportional to a parameter which measures the heterogeneity of the update rules. Interestingly, the latter does not necessarily increase with the mean number of discrete states per node. Still, allowing for multiple states decreases the critical connectivity as compared to random Boolean networks, and thus leads to biologically unrealistic situations. Therefore, we study two approaches to increase the critical connectivity.First, we demonstrate that each network can be kept in its frozen regime by sufficiently biasing the update rules. Second, we restrict the randomly chosen update rules to a subclass of biologically more meaningful functions. These functions are characterized based on a thermodynamic model of gene regu- Preprint submitted to Journal of Theoretical BiologyMay 27, 2010 lation. We analytically show that their usage indeed increases the critical connectivity. From a general point of view, our thermodynamic considerations link discrete and continuous models of gene regulatory networks.

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The phase diagram of random threshold networks

Cited by 36 publications

References 13 publications

Influence and Dynamic Behavior in Random Boolean Networks

Influence and Dynamic Behavior in Random Boolean Networks

Dynamical regimes in non-ergodic random Boolean networks

Biologically meaningful update rules increase the critical connectivity of generalized Kauffman networks

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