Numerical and Theoretical Studies of Noise Effects in the Kauffman Model

In this work we analyze the stochastic dynamics of the Kauffman model evolving under the influence of noise. By considering the average crossing time between two distinct trajectories, we show that different Kauffman models exhibit a similar kind of behavior, even when the structure of their basins of attraction is quite different. This can be considered as a robust property of these models. We present numerical results for the full range of noise level and obtain approximate analytic expressions for the above crossing time as a function of the noise in the limit cases of small and large noise levels.Comment: 24 pages, 9 figures, Submitted to the Journal of Statistical Physic

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

Facies recognition using wavelet based fractal analysis and waveform classifier at the Oritupano-A Field, Venezuela

We have used a Wavelet Based Fractal Analysis (WBFA) and a Waveform Classifier (WC) to recognize lithofacies at the Oritupano A field (Oritupano-Leona Block, Venezuela). The WBFA was applied first to Sonic, Density, Gamma Ray and Porosity well logs in the area. The logs that give the best response to the WBFA are the Gamma Ray and NPHI (porosity) logs. In the case of the logs, the lithological content could be associated to the fractal parameters: slope, intercept and fractal dimension. The map obtained using the fractal dimension shows tendencies that generally agree with the depositional patterns previously observed in conventional geological maps. According to the results obtained in this study, zones with fractal dimension values lower than 0.9 correspond to sandstone channels. Values between 0.9 and 1.2 coincide with the interdistributary deltaic shelf and values greater than 1.2 might be associated with zones of greater shale content. The WBFA and WC results obtained for the seismic data show no relation with the lithofacies. The lost of low and high frequencies in these seismic data, as well as phase problems, could be the reasons for this behavior

Phase transitions in systems of self-propelled agents and related network models

An important characteristic of flocks of birds, school of fish, and many similar assemblies of self-propelled particles is the emergence of states of collective order in which the particles move in the same direction. When noise is added into the system, the onset of such collective order occurs through a dynamical phase transition controlled by the noise intensity. While originally thought to be continuous, the phase transition has been claimed to be discontinuous on the basis of recently reported numerical evidence. We address this issue by analyzing two representative network models closely related to systems of self-propelled particles. We present analytical as well as numerical results showing that the nature of the phase transition depends crucially on the way in which noise is introduced into the system.Comment: Four pages, four figures. Submitted to PR

Bootstrap testing for cross-correlation under low firing activity

A new cross-correlation synchrony index for neural activity is proposed. The index is based on the integration of the kernel estimation of the cross-correlation function. It is used to test for the dynamic synchronization levels of spontaneous neural activity under two induced brain states: sleep-like and awake-like. Two bootstrap resampling plans are proposed to approximate the distribution of the test statistics. The results of the first bootstrap method indicate that it is useful to discern significant differences in the synchronization dynamics of brain states characterized by a neural activity with low firing rate. The second bootstrap method is useful to unveil subtle differences in the synchronization levels of the awake-like state, depending on the activation pathway.Comment: 22 pages, 7 figure

Stable and unstable attractors in Boolean networks

Boolean networks at the critical point have been a matter of debate for many years as, e.g., scaling of number of attractor with system size. Recently it was found that this number scales superpolynomially with system size, contrary to a common earlier expectation of sublinear scaling. We here point to the fact that these results are obtained using deterministic parallel update, where a large fraction of attractors in fact are an artifact of the updating scheme. This limits the significance of these results for biological systems where noise is omnipresent. We here take a fresh look at attractors in Boolean networks with the original motivation of simplified models for biological systems in mind. We test stability of attractors w.r.t. infinitesimal deviations from synchronous update and find that most attractors found under parallel update are artifacts arising from the synchronous clocking mode. The remaining fraction of attractors are stable against fluctuating response delays. For this subset of stable attractors we observe sublinear scaling of the number of attractors with system size.Comment: extended version, additional figur

The properties of attractors of canalyzing random Boolean networks

We study critical random Boolean networks with two inputs per node that contain only canalyzing functions. We present a phenomenological theory that explains how a frozen core of nodes that are frozen on all attractors arises. This theory leads to an intuitive understanding of the system's dynamics as it demonstrates the analogy between standard random Boolean networks and networks with canalyzing functions only. It reproduces correctly the scaling of the number of nonfrozen nodes with system size. We then investigate numerically attractor lengths and numbers, and explain the findings in terms of the properties of relevant components. In particular we show that canalyzing networks can contain very long attractors, albeit they occur less often than in standard networks.Comment: 9 pages, 8 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

Phase transition in a class of non-linear random networks

We discuss the complex dynamics of a non-linear random networks model, as a function of the connectivity k between the elements of the network. We show that this class of networks exhibit an order-chaos phase transition for a critical connectivity k = 2. Also, we show that both, pairwise correlation and complexity measures are maximized in dynamically critical networks. These results are in good agreement with the previously reported studies on random Boolean networks and random threshold networks, and show once again that critical networks provide an optimal coordination of diverse behavior.Comment: 9 pages, 3 figures, revised versio

Static Pairwise Annihilation in Complex Networks

We study static annihilation on complex networks, in which pairs of connected particles annihilate at a constant rate during time. Through a mean-field formalism, we compute the temporal evolution of the distribution of surviving sites with an arbitrary number of connections. This general formalism, which is exact for disordered networks, is applied to Kronecker, Erd\"os-R\'enyi (i.e. Poisson) and scale-free networks. We compare our theoretical results with extensive numerical simulations obtaining excellent agreement. Although the mean-field approach applies in an exact way neither to ordered lattices nor to small-world networks, it qualitatively describes the annihilation dynamics in such structures. Our results indicate that the higher the connectivity of a given network element, the faster it annihilates. This fact has dramatic consequences in scale-free networks, for which, once the ``hubs'' have been annihilated, the network disintegrates and only isolated sites are left.Comment: 7 Figures, 10 page