Phase Transitions In Random Boolean Networks With Different Updating Schemes (original) (raw)

The information dynamics of phase transitions in random Boolean networks

2008

Abstract Random Boolean Networks (RBNs) are discrete dynamical systems which have been used to model Gene Regulatory Networks. We investigate the well-known phase transition between ordered and chaotic behavior in RBNs from the perspective of the distributed computation conducted by their nodes. We use a recently published framework to characterize the distributed computation in terms of its underlying information dynamics: information storage, information transfer and information modification.

On the dynamics of random Boolean networks with scale-free outgoing connections

Physica A-statistical Mechanics and Its Applications, 2004

In the classical model of Random Boolean Networks (RBN) the number of incoming connections is the same for every node, while the distribution of outgoing links is Poissonian. These RBN are known to display two major dynamical behaviours, depending upon the value of some model parameters: an "ordered" and a "chaotic" regime. We introduce a modiÿcation of the classical way of building a RBN, which maintains the property that all the nodes have the same number of incoming links, but which gives rise to a scale-free distribution of outgoing connections. Because of this modiÿcation, the dynamical properties are deeply modiÿed: the number of attractors is much smaller than in classical RBN, their length and the duration of the transients are shorter. Moreover, the number of di erent attractors is almost independent of the network size, over almost three orders of magnitudes (while in classical RBN this number grows with the size of the network). These results are based upon a detailed study of networks where each node has two input connections. A limited study of networks with three input connections per node shows that also in this case the number of attractors is almost independent of the network size.

Random Boolean network model exhibiting deterministic chaos

Physical Review E, 2004

This paper considers a simple Boolean network with N nodes, each node's state at time t being determined by a certain number of parent nodes, which may vary from one node to another. This is an extension of a model studied by Andrecut and Ali [Int. J. Mod. Phys. B 15, 17 (2001)], who consider the same number of parents for all nodes. We make use of the same Boolean rule as Andrecut and Ali, provide a generalization of the formula for the probability of finding a node in state 1 at a time t, and use simulation methods to generate consecutive states of the network for both the real system and the model. The results match well. We study the dynamics of the model through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed point analysis. We show that the route to chaos is due to a cascade of period-doubling bifurcations which turn into reversed (period-halving) bifurcations for certain combinations of parameter values.

Asynchronous dynamics of random Boolean networks

IEEE International Conference on Neural Networks, 1988

The undersigned hereby certify that they have read and recommend to the Faculty of Graduate Studies for acceptance a thesis entitled "Dynamics of Random Boolean Networks Governed by a Generalization of Rule 22 of Elementary Cellular Automata" by Gary L. Beck in partial fulfillment of the requirements for the degree

Attractor and basin entropies of random Boolean networks under asynchronous stochastic update

Physical Review E, 2010

We introduce a numerical method to study random Boolean networks with asynchronous stochastic update. Each node in the network of states starts with equal occupation probability and this probability distribution then evolves to a steady state. Nodes left with finite occupation probability determine the attractors and the sizes of their basins. As for synchronous update, the basin entropy grows with system size only for critical networks, where the distribution of attractor lengths is a power law. We determine analytically the distribution for the number of attractors and basin sizes for frozen networks with connectivity K = 1.

Order Parameters, Lyapunov Exponents, and Control in Random Boolean Networks

Working Papers, 1997

A new order parameter approximation to Random Boolean Networks (RBN) is introduced, based on the concept of Boolean derivative. A statistical argument involving an annealed approximation is used, allowing to measure the order parameter in terms of the statistical properties of a random matrix. Using the same formalism, a Lyapunov exponent is calculated, allowing to provide the onset of damage spreading through the network and how sensitive it is to single ips. Finnally, these measures are used in order to estimate the critical boundaries for chaos control in RBN.

Phase transition in random networks with multiple states

Arxiv preprint adap-org/9907011, 1999

The critical boundaries separating ordered from chaotic behavior in randomly wired S-state networks are calculated. These networks are a natural generalization of random Boolean nets and are proposed as on extended approach to genetic regulatory systems, sets of cells in different states or collectives of agents engaged into a set of S possible tasks. A order parameter for the transition is computed and analysed. The relevance of these networks to biology, their relationships with standard cellular automata and possible extensions are outlined.

Robustness and Information Propagation in Attractors of Random Boolean Networks

PLoS ONE, 2012

Attractors represent the long-term behaviors of Random Boolean Networks. We study how the amount of information propagated between the nodes when on an attractor, as quantified by the average pairwise mutual information (I A ), relates to the robustness of the attractor to perturbations (R A ). We find that the dynamical regime of the network affects the relationship between I A and R A . In the ordered and chaotic regimes, I A is anti-correlated with R A , implying that attractors that are highly robust to perturbations have necessarily limited information propagation. Between order and chaos (for socalled ''critical'' networks) these quantities are uncorrelated. Finite size effects cause this behavior to be visible for a range of networks, from having a sensitivity of 1 to the point where I A is maximized. In this region, the two quantities are weakly correlated and attractors can be almost arbitrarily robust to perturbations without restricting the propagation of information in the network.

Propagation of external regulation and asynchronous dynamics in random Boolean networks

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2007

Boolean Networks and their dynamics are of great interest as abstract modeling schemes in various disciplines, ranging from biology to computer science. Whereas parallel update schemes have been studied extensively in past years, the level of understanding of asynchronous updates schemes is still very poor. In this paper we study the propagation of external information given by regulatory input variables into a random Boolean network. We compute both analytically and numerically the time evolution and the asymptotic behavior of this propagation of external regulation (PER). In particular, this allows us to identify variables which are completely determined by this external information. All those variables in the network which are not directly fixed by PER form a core which contains in particular all non-trivial feedback loops. We design a message-passing approach allowing to characterize the statistical properties of these cores in dependence of the Boolean network and the external condition. At the end we establish a link between PER dynamics and the full random asynchronous dynamics of a Boolean network.