Joern Davidsen

Extreme value statistics and return intervals in long-range correlated uniform deviates.

We study extremal statistics and return intervals in stationary long-range correlated sequences for which the underlying probability density function is bounded and uniform. The extremal statistics we consider (e.g., maximum relative to minimum) are such that the reference point from which the maximum is measured is itself a random quantity. We analytically calculate the limiting distributions for independent and identically distributed random variables, and use these as a reference point for correlated cases. The distributions are different from that of the maximum itself (i.e., a Weibull distribution), reflecting the fact that the distribution of the reference point either dominates over or convolves with the distribution of the maximum. The functional form of the limiting distributions is unaffected by correlations, although the convergence is slower. We show that our findings can be directly generalized to a wide class of stochastic processes. We also analyze return interval distributions, and compare them to recent conjectures of their functional form.

Nonlinear Geophysics: Why We Need It

Few geoscientists would deny that effects are often sensitively dependent on causes, or that their amplification is commonly so strong as to give rise to qualitatively new “emergent” properties, or that geostructures are typically embedded one within another in a hierarchy. Starting in the 1980s, a growing number felt the need to underline the absolute importance of such nonlinearity through workshops and conferences. Building on this, the European Geosciences Union (EGU) organized a nonlinear processes (NP) section in 1990; AGU established a nonlinear geophysics (NG) focus group in 1997; and both unions began collaborating on an academic journal, Nonlinear Processes in Geophysics, in 1994.

Network of recurrent events for the Olami-Feder-Christensen model.

We numerically study the dynamics of a discrete spring-block model introduced by Olami, Feder, and Christensen (OFC) to mimic earthquakes and investigate to what extent this simple model is able to reproduce the observed spatiotemporal clustering of seismicity. Following a recently proposed method to characterize such clustering by networks of recurrent events [J. Davidsen, P. Grassberger, and M. Paczuski, Geophys. Res. Lett. 33, L11304 (2006)], we find that for synthetic catalogs generated by the OFC model these networks have many nontrivial statistical properties. This includes characteristic degree distributions, very similar to what has been observed for real seismicity. There are, however, also significant differences between the OFC model and earthquake catalogs, indicating that this simple model is insufficient to account for certain aspects of the spatiotemporal clustering of seismicity.

Record-breaking avalanches in driven threshold systems

Record-breaking avalanches generated by the dynamics of several driven nonlinear threshold models are studied. Such systems are characterized by intermittent behavior, where a slow buildup of energy is punctuated by an abrupt release of energy through avalanche events, which usually follow scale-invariant statistics. From the simulations of these systems it is possible to extract sequences of record-breaking avalanches, where each subsequent record-breaking event is larger in magnitude than all previous events. In the present work, several cellular automata are analyzed, among them the sandpile model, the Manna model, the Olami-Feder-Christensen (OFC) model, and the forest-fire model to investigate the record-breaking statistics of model avalanches that exhibit temporal and spatial correlations. Several statistical measures of record-breaking events are derived analytically and confirmed through numerical simulations. The statistics of record-breaking avalanches for the four models are compared to those of record-breaking events extracted from the sequences of independent and identically distributed (i.i.d.) random variables. It is found that the statistics of record-breaking avalanches for the above cellular automata exhibit behavior different from that observed for i.i.d. random variables, which in turn can be used to characterize complex spatiotemporal dynamics. The most pronounced deviations are observed in the case of the OFC model with a strong dependence on the conservation parameter of the model. This indicates that avalanches in the OFC model are not independent and exhibit spatiotemporal correlations.

Random sampling versus exact enumeration of attractors in random Boolean networks

We clarify the effect different sampling methods and weighting schemes have on the statistics of attractors in ensembles of random Boolean networks (RBNs). We directly measure the cycle lengths of attractors and the sizes of basins of attraction in RBNs using exact enumeration of the state space. In general, the distribution of attractor lengths differs markedly from that obtained by randomly choosing an initial state and following the dynamics to reach an attractor. Our results indicate that the former distribution decays as a power law with exponent 1 for all connectivities K>1 in the infinite system size limit. In contrast, the latter distribution decays as a power law only for K=2. This is because the mean basin size grows linearly with the attractor cycle length for K>2, and is statistically independent of the cycle length for K=2. We also find that the histograms of basin sizes are strongly peaked at integer multiples of powers of two for K<3.

Avalanches, branching ratios, and clustering of attractors in random Boolean networks and in the segment polarity network of Drosophila

We discuss basic features of emergent complexity in dynamical systems far from equilibrium by focusing on the network structure of their state space. We start by measuring the distributions of avalanche and transient times in random Boolean networks (RBNs) and in the Drosophila polarity network by exact enumeration. A transient time is the duration of the transient from a starting state to an attractor. An avalanche is a special transient which starts as a single Boolean element perturbation of an attractor state. Significant differences at short times between the avalanche and the transient times for RBNs with small connectivity K—compared to the number of elements N—indicate that attractors tend to cluster in configuration space. In addition, one bit flip has a non-negligible chance to put an attractor state directly onto another attractor. This clustering is also present in the segment polarity gene network of Drosophila melanogaster, suggesting that this may be a robust feature of biological regulatory networks. We also define and measure a branching ratio for the state space networks and find evidence for a new timescale that diverges roughly linearly with N for 2≤KN. Analytic arguments show that this timescale does not appear in the random map nor can the random map exhibit clustering of attractors. We further show that for K=2 the branching ratio exhibits the largest variation with distance from the attractor compared to other values of K and that the avalanche durations exhibit no characteristic scale within our statistical resolution. Hence, we propose that the branching ratio and the avalanche duration are new indicators for scale-free behavior that may or may not be found simultaneously with other indicators of emergent complexity in extended, deterministic dynamical systems.