Relevant components in critical random Boolean networks

Random Boolean networks were introduced in 1969 by Kauffman as a model for gene regulation. By combining analytical arguments and efficient numerical simulations, we evaluate the properties of relevant components of critical random Boolean networks independently of update scheme. As known from previous work, the number of relevant components grows logarithmically with network size. We find that in most networks all relevant nodes with more than one relevant input sit in the same component, while all other relevant components are simple loops. As the proportion of nonfrozen nodes with two relevant inputs increases, the number of relevant components decreases and the size and complexity of the largest complex component grows. We evaluate the probability distribution of different types of complex components in an ensemble of networks and confirm that it becomes independent of network size in the limit of large network size. In this limit, we determine analytically the frequencies of occurence of complex components with different topologies.Comment: 9 pages, 6 figure

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

Formation of Space-Time Structure in a Forest-Fire Model

We present a general stochastic forest-fire model which shows a variety of different structures depending on the parameter values. The model contains three possible states per site (tree, burning tree, empty site) and three parameters (tree growth probability $p$, lightning probability $f$, and immunity $g$). We review analytic and computer simulation results for a quasideterministic state with spiral-shaped fire fronts, for a percolation-like phase transition and a self-organized critical state. Possible applications to excitable systems are discussed.Comment: 20 pages REVTEX, 9 figures upon reques

Modelling Food Webs

We review theoretical approaches to the understanding of food webs. After an overview of the available food web data, we discuss three different classes of models. The first class comprise static models, which assign links between species according to some simple rule. The second class are dynamical models, which include the population dynamics of several interacting species. We focus on the question of the stability of such webs. The third class are species assembly models and evolutionary models, which build webs starting from a few species by adding new species through a process of "invasion" (assembly models) or "speciation" (evolutionary models). Evolutionary models are found to be capable of building large stable webs.Comment: 34 pages, 2 figures. To be published in "Handbook of graphs and networks" S. Bornholdt and H. G. Schuster (eds) (Wiley-VCH, Berlin

On the influence of the intermolecular potential on the wetting properties of water on silica surfaces

We study the wetting properties of water on silica surfaces using molecular dynamics (MD) simulations. To describe the intermolecular interaction between water and silica atoms, two types of interaction potential models are used: the standard Br\'odka and Zerda (BZ) model, and the Gulmen and Thompson (GT) model. We perform an in-depth analysis of the influence of the choice of the potential on the arrangement of the water molecules in partially filled pores and on top of silica slabs. We find that at moderate pore filling ratios, the GT silica surface is completely wetted by water molecules, which agrees well with experimental findings, while the commonly used BZ surface is less hydrophilic and is only partially wetted. We interpret our simulation results using an analytical calculation of the phase diagram of water in partially filled pores. Moreover, an evaluation of the contact angle of the water droplet on top of the silica slab reveals that the interaction becomes more hydrophilic with increasing slab thickness and saturates around 2.5-3 nm, in agreement with the experimentally found value. Our analysis also shows that the hydroaffinity of the surface is mainly determined by the electrostatic interaction, but that the van der Waals interaction nevertheless is strong enough that it can turn a hydrophobic surface into a hydrophilic surface.Comment: Article: 9 pages, 7 Figures. There is also a supplementary information file: 2 pages, 3 Figure

Scaling laws and simulation results for the self--organized critical forest--fire model

We discuss the properties of a self--organized critical forest--fire model which has been introduced recently. We derive scaling laws and define critical exponents. The values of these critical exponents are determined by computer simulations in 1 to 8 dimensions. The simulations suggest a critical dimension $d_c=6$ above which the critical exponents assume their mean--field values. Changing the lattice symmetry and allowing trees to be immune against fire, we show that the critical exponents are universal.Comment: 12 pages, postscript uuencoded, figures included, to appear in Phys. Rev.

Boolean networks with reliable dynamics

We investigated the properties of Boolean networks that follow a given reliable trajectory in state space. A reliable trajectory is defined as a sequence of states which is independent of the order in which the nodes are updated. We explored numerically the topology, the update functions, and the state space structure of these networks, which we constructed using a minimum number of links and the simplest update functions. We found that the clustering coefficient is larger than in random networks, and that the probability distribution of three-node motifs is similar to that found in gene regulation networks. Among the update functions, only a subset of all possible functions occur, and they can be classified according to their probability. More homogeneous functions occur more often, leading to a dominance of canalyzing functions. Finally, we studied the entire state space of the networks. We observed that with increasing systems size, fixed points become more dominant, moving the networks close to the frozen phase.Comment: 11 Pages, 15 figure

The phase diagram of random threshold networks

Threshold networks are used as models for neural or gene regulatory networks. They show a rich dynamical behaviour with a transition between a frozen and a chaotic phase. We investigate the phase diagram of randomly connected threshold networks with real-valued thresholds h and a fixed number of inputs per node. The nodes are updated according to the same rules as in a model of the cell-cycle network of Saccharomyces cereviseae [PNAS 101, 4781 (2004)]. Using the annealed approximation, we derive expressions for the time evolution of the proportion of nodes in the "on" and "off" state, and for the sensitivity $\lambda$. The results are compared with simulations of quenched networks. We find that for integer values of h the simulations show marked deviations from the annealed approximation even for large networks. This can be attributed to the particular choice of the updating rule.Comment: 8 pages, 6 figure

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

On the properties of cycles of simple Boolean networks

We study two types of simple Boolean networks, namely two loops with a cross-link and one loop with an additional internal link. Such networks occur as relevant components of critical K=2 Kauffman networks. We determine mostly analytically the numbers and lengths of cycles of these networks and find many of the features that have been observed in Kauffman networks. In particular, the mean number and length of cycles can diverge faster than any power law.Comment: 10 pages, 8 figure