Stability of the Kauffman Model

Random Boolean networks, the Kauffman model, are revisited by means of a novel decimation algorithm, which reduces the networks to their dynamical cores. The average size of the removed part, the stable core, grows approximately linearly with N, the number of nodes in the original networks. We show that this can be understood as the percolation of the stability signal in the network. The stability of the dynamical core is investigated and it is shown that this core lacks the well known stability observed in full Kauffman networks. We conclude that, somewhat counter-intuitive, the remarkable stability of Kauffman networks is generated by the dynamics of the stable core. The decimation method is also used to simulate large critical Kauffman networks. For networks up to N=32 we perform full enumeration studies. Strong evidence is provided for that the number of limit cycles grows linearly with N. This result is in sharp contrast to the often cited $\sqrt{N}$ behavior.Comment: 12 pages, 4 figure

The Strong-Coupling Expansion in Simplicial Quantum Gravity

We construct the strong-coupling series in 4d simplicial quantum gravity up to volume 38. It is used to calculate estimates for the string susceptibility exponent gamma for various modifications of the theory. It provides a very efficient way to get a first view of the phase structure of the models.Comment: LATTICE98(surfaces), 3 pages, 4 eps figure

Probabilistic estimation of microarray data reliability and underlying gene expression

Background: The availability of high throughput methods for measurement of mRNA concentrations makes the reliability of conclusions drawn from the data and global quality control of samples and hybridization important issues. We address these issues by an information theoretic approach, applied to discretized expression values in replicated gene expression data. Results: Our approach yields a quantitative measure of two important parameter classes: First, the probability $P(\sigma | S)$ that a gene is in the biological state $\sigma$ in a certain variety, given its observed expression $S$ in the samples of that variety. Second, sample specific error probabilities which serve as consistency indicators of the measured samples of each variety. The method and its limitations are tested on gene expression data for developing murine B-cells and a $t$-test is used as reference. On a set of known genes it performs better than the $t$-test despite the crude discretization into only two expression levels. The consistency indicators, i.e. the error probabilities, correlate well with variations in the biological material and thus prove efficient. Conclusions: The proposed method is effective in determining differential gene expression and sample reliability in replicated microarray data. Already at two discrete expression levels in each sample, it gives a good explanation of the data and is comparable to standard techniques.Comment: 11 pages, 4 figure

Phase Transition of 4D Simplicial Quantum Gravity with U(1) Gauge Field

The phase transition of 4D simplicial quantum gravity coupled to U(1) gauge fields is studied using Monte-Carlo simulations. The phase transition of the dynamical triangulation model with vector field ($N_{V}=1$) is smooth as compared with the pure gravity($N_{V}=0$). The node susceptibility ($\chi$) is studied in the finite size scaling method. At the critical point, the node distribution has a sharp peak in contrast to the double peak in the pure gravity. From the numerical results, we expect that 4D simplicial quantum gravity with U(1) vector fields has higher order phase transition than 1st order, which means the possibility to take the continuum limit at the critical point.Comment: 3 pages, latex, 3 eps figures, uses espcrc2.sty. Talk presented at LATTICE99(gravity

On the number of attractors in random Boolean networks

The evaluation of the number of attractors in Kauffman networks by Samuelsson and Troein is generalized to critical networks with one input per node and to networks with two inputs per node and different probability distributions for update functions. A connection is made between the terms occurring in the calculation and between the more graphic concepts of frozen, nonfrozen and relevant nodes, and relevant components. Based on this understanding, a phenomenological argument is given that reproduces the dependence of the attractor numbers on system size.Comment: 6 page

Universality of hypercubic random surfaces

We study universality properties of the Weingarten hyper-cubic random surfaces. Since a long time ago the model with a local restriction forbidding surface self-bendings has been thought to be in a different universality class from the unrestricted model defined on the full set of surfaces. We show that both models in fact belong to the same universality class with the entropy exponent gamma = 1/2 and differ by finite size effects which are much more pronounced in the restricted model.Comment: 8 pages, 3 figure

Phase transition and topology in 4d simplicial gravity

We present data indicating that the recent evidence for the phase transition being of first order does not result from a breakdown of the ergodicity of the algorithm. We also present data showing that the thermodynamical limit of the model is independent of topology.Comment: 3 latex pages + 4 ps fig. + espcrc2.sty. Talk presented at LATTICE(gravity

Simulating Four-Dimensional Simplicial Gravity using Degenerate Triangulations

We extend a model of four-dimensional simplicial quantum gravity to include degenerate triangulations in addition to combinatorial triangulations traditionally used. Relaxing the constraint that every 4-simplex is uniquely defined by a set of five distinct vertexes, we allow triangulations containing multiply connected simplexes and distinct simplexes defined by the same set of vertexes. We demonstrate numerically that including degenerated triangulations substantially reduces the finite-size effects in the model. In particular, we provide a strong numerical evidence for an exponential bound on the entropic growth of the ensemble of degenerate triangulations, and show that a discontinuous crumpling transition is already observed on triangulations of volume N_4 ~= 4000.Comment: Latex, 8 pages, 4 eps-figure

4d Simplicial Quantum Gravity Interacting with Gauge Matter Fields

The effect of coupling non-compact $U(1)$ gauge fields to four dimensional simplicial quantum gravity is studied using strong coupling expansions and Monte Carlo simulations. For one gauge field the back-reaction of the matter on the geometry is weak. This changes, however, as more matter fields are introduced. For more than two gauge fields the degeneracy of random manifolds into branched polymers does not occur, and the branched polymer phase seems to be replaced by a new phase with a negative string susceptibility exponent $\gamma$ and fractal dimension $d_H \approx 4$.Comment: latex2e, 10 pages incorporating 2 tables and 3 figures (using epsf