Critical Boolean networks with scale-free in-degree distribution

We investigate analytically and numerically the dynamical properties of critical Boolean networks with power-law in-degree distributions. When the exponent of the in-degree distribution is larger than 3, we obtain results equivalent to those obtained for networks with fixed in-degree, e.g., the number of the non-frozen nodes scales as $N^{2/3}$ with the system size $N$. When the exponent of the distribution is between 2 and 3, the number of the non-frozen nodes increases as $N^x$, with $x$ being between 0 and 2/3 and depending on the exponent and on the cutoff of the in-degree distribution. These and ensuing results explain various findings obtained earlier by computer simulations.Comment: 5 pages, 1 graph, 1 sketch, submitte

Boolean networks as modeling framework

In a network, the components of a given system are represented as nodes, the interactions are abstracted as links between the nodes. Boolean networks refer to a class of dynamics on networks, in fact it is the simplest possible dynamics where each node has a value 0 or 1. This allows to investigate extensively the dynamics both analytically and by numerical experiments. The present article focuses on the theoretical concept of relevant components and their immediate application in plant biology. References for more in-depth treatment of the mathematical details are also given

Historical variability of the density stratification in the Laptev Sea

The Laptev Sea is a key region for sea ice formation and export to the inner Arctic. During winter, wind-ice-dynamics repeatedly produce open water areas (polynyas) with extensive heat fluxes, sea ice formation and water mass modification. In summer, the oceanic processes are strongly influenced by the enormous freshwater discharge of Siberian rivers. All this influences the density distribution. We therefore present a comprehensive analysis of the historical temperature and salinity data which is episodically available since around 1910. The variability of the density stratification is important for mixing across the water column

Number and length of attractors in a critical Kauffman model with connectivity one

The Kauffman model describes a system of randomly connected nodes with dynamics based on Boolean update functions. Though it is a simple model, it exhibits very complex behavior for "critical" parameter values at the boundary between a frozen and a disordered phase, and is therefore used for studies of real network problems. We prove here that the mean number and mean length of attractors in critical random Boolean networks with connectivity one both increase faster than any power law with network size. We derive these results by generating the networks through a growth process and by calculating lower bounds.Comment: 4 pages, no figure, no table; published in PR

Attractor and Basin Entropies of Random Boolean Networks Under Asynchronous Stochastic Update

We introduce a numerical method to study random Boolean networks with asynchronous stochas- tic 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.Comment: 5 pages, 3 figures, in submissio

The dynamics of critical Kauffman networks under asynchronous stochastic update

We show that the mean number of attractors in a critical Boolean network under asynchronous stochastic update grows like a power law and that the mean size of the attractors increases as a stretched exponential with the system size. This is in strong contrast to the synchronous case, where the number of attractors grows faster than any power law.Comment: submitted to PR

Critical Kauffman networks under deterministic asynchronous update

We investigate the influence of a deterministic but non-synchronous update on Random Boolean Networks, with a focus on critical networks. Knowing that ``relevant components'' determine the number and length of attractors, we focus on such relevant components and calculate how the length and number of attractors on these components are modified by delays at one or more nodes. The main findings are that attractors decrease in number when there are more delays, and that periods may become very long when delays are not integer multiples of the basic update step.Comment: 8 pages, 3 figures, submitted to a journa