Self-organized Critical Behavior Research Articles

Effects of Vertex Activity and Self-organized Criticality Behavior on a Weighted Evolving Network

Effects of vertex activity have been analyzed on a weighted evolving network. The network is characterized by the probability distribution of vertex strength, each edge weight and evolution of the strength of vertices with different vertex activities. The model exhibits self-organized criticality behavior. The probability distribution of avalanche size for different network sizes is also shown. In addition, there is a power law relation between the size and the duration of an avalanche and the average of avalanche size has been studied for different vertex activities.

Emergence of complexity in evolving niche-model food webs

We have analysed mechanisms that promote the emergence of complex structures in evolving model food webs. The niche model is used to determine predator–prey relationships. Complexity is measured by species richness as well as trophic level structure and link density. Adaptive dynamics that allow predators to concentrate on the prey species they are best adapted to lead to a strong increase in species number but have only a small effect on the number and relative occupancy of trophic levels. The density of active links also remains small but a high number of potential links allows the network to adjust to changes in the species composition (emergence and extinction of species). Incorporating effects of body size on individual metabolism leads to a more complex trophic level structure: both the maximum and the average trophic level increase. So does the density of active links. Taking body size effects into consideration does not have a measurable influence on species richness. If species are allowed to adjust their foraging behaviour, the complexity of the evolving networks can also be influenced by the size of the external resources. The larger the resources, the larger and more complex is the food web it can sustain. Body size effects and increasing resources do not change size and the simple structure of the evolving networks if adaptive foraging is prohibited. This leads to the conclusion that in the framework of the niche model adaptive foraging is a necessary but not sufficient condition for the emergence of complex networks. It is found that despite the stabilising effect of foraging adaptation the system displays elements of self-organised critical behaviour.

Dynamical synapses causing self-organized criticality in neural networks

Self-organized criticality1 is one of the key concepts to describe the emergence of complexity in natural systems. The concept asserts that a system self-organizes into a critical state where system observables are distributed according to a power law. Prominent examples of self-organized critical dynamics include piling of granular media2, plate tectonics3 and stick–slip motion4. Critical behaviour has been shown to bring about optimal computational capabilities5, optimal transmission6, storage of information7 and sensitivity to sensory stimuli8,9,10. In neuronal systems, the existence of critical avalanches was predicted11 and later observed experimentally6,12,13. However, whereas in the experiments generic critical avalanches were found, in the model of ref. 11 they only show up if the set of parameters is fine-tuned externally to a critical transition state. Here, we demonstrate analytically and numerically that by assuming (biologically more realistic) dynamical synapses14 in a spiking neural network, the neuronal avalanches turn from an exceptional phenomenon into a typical and robust self-organized critical behaviour, if the total resources of neurotransmitter are sufficiently large.

Quantifying fluvial non linearity and finding self organized criticality? Insights from simulations of river basin evolution

A numerical study was undertaken to investigate non linearity and the potential for self-organized criticality (SOC) in the evolution of river basins. Twenty-three simulations were carried out, using the authors' CAESAR landscape evolution model, in which the magnitude of storm events, variability of storm events, sediment heterogeneity, sources of sediment supply, and catchment morphology are systematically varied to evaluate their importance as possible drivers for non linear behavior and SOC. Temporal fluctuations in simulated sediment yield show notable non linear behavior. Storm magnitude and occurrence of landslides appear to have little impact on variability of the sediment yield, when compared to the impacts of sediment heterogeneity, rainfall variability and catchment morphology. Particularly, it appears that the non linearity of sediment yields results from the manner in which the catchment processes the variable rainfall, rather than just the rainfall variability itself. The variations in sediment yield show a power law magnitude–frequency distribution, which is a possible, but inconclusive, indicator of SOC. However, several other, more qualitative arguments can be made to support the case for SOC in these simulations. Specifically, we identify the nature of the critical state and suggest two cascade mechanisms by which the system can organize itself around this critical state. Combined, these arguments indicate that simulated evolution of river basins indeed exhibits SOC, at least with respect to sediment yield. The critical state appears to be an indicator of the connectivity of the drainage network. Thus, the simulations indicate that, unlike traditional SOC systems, the critical state of the system can vary in time, as sudden changes in drainage network connectivity may result in sudden changes in the SOC behavior of the system.

Applying multifractality and the self‐organized criticality theory to describe the temporal rainfall regimes in Andalusia (southern Spain)

AbstractA multifractal analysis of hourly and daily rainfall data recorded at four locations of Andalusia (southern Spain) was carried out in order to study the temporal structure of rainfall and to find differences between both time resolutions. The results show that an algebraic tail is required to fit the probability distribution of the extreme rain events for all the cases. The presence of a multifractal phase transition associated with a critical moment in the empirical moments scaling exponent function was also detected. Both facts indicate that the rainfall process is a case of self‐organized criticality (SOC) dynamics, although the results differ for each place according to the time resolution and the nature of the rainfall, either convective or frontal. This SOC behaviour is related to a statistically steady state that implies the presence of clusterization in the time‐occurrence sequence of rain events. Such fluctuations have been shown by performing the analysis of the Fano and Allan factors and the count‐based periodogram. The values for the “synoptic maximum”, the typical lifetime of planetary scale atmospheric structures, have been obtained for each place and some important periodicities have been detected when dealing with extremes. Copyright © 2007 John Wiley & Sons, Ltd.

Scaling properties in forest fire sequences: The human role in the order of nature

Since the 1990s, forest-fire models focused on self-organized criticality behaviour on both spatial and temporal scales. Recent related studies have suggested that the spatial fractality of wildfires is due to the heterogeneous distribution of forest mass (frequency–size distribution fitted to a power-law). However, the differences observed between real data and simulated data, due to the multiplicity of interacting factors, many of them unknown, remain unclear. As the great majority of forest fires are caused by humans, in the present work, we have studied spatial and temporal distributions of forest-fire sequences detected in the eastern part of Spain, depending on the wildfire origin. We showed how humans modify spatial and temporal scale invariant properties, playing a role in the order of nature. This human impact, especially, has important implications: decreasing the inter-event interval and increasing the “sparking frequency” on forest-fire modelling.

Critical dynamic approach to stationary states in complex systems

A dynamic scaling Ansatz for the approach to stationary states in complex systems is proposed and tested by means of extensive simulations applied to both the Bak-Sneppen (BS) model, which exhibits robust Self-Organised Critical (SOC) behaviour, and the Game of Life (GOL) of J. Conway, whose critical behaviour is under debate. Considering the dynamic scaling behaviour of the density of sites (ρ(t)), it is shown that i) by starting the dynamic measurements with configurations such that ρ(t=0) →0, one observes an initial increase of the density with exponents θ= 0.12(2) and θ= 0.11(2) for the BS and GOL models, respectively; ii) by using initial configurations with ρ(t=0) →1, the density decays with exponents δ= 0.47(2) and δ= 0.28(2) for the BS and GOL models, respectively. It is also shown that the temporal autocorrelation decays with exponents Ca = 0.35(2) (Ca = 0.35(5)) for the BS (GOL) model. By using these dynamically determined critical exponents and suitable scaling relationships, we also obtain the dynamic exponents z = 2.10(5) (z = 2.10(5)) for the BS (GOL) model. Based on this evidence we conclude that the dynamic approach to stationary states of the investigated models can be described by suitable power-law functions of time with well-defined exponents.

Realization of SOC behavior in a dc glow discharge plasma

Experimental observations consistent with Self Organized Criticality (SOC) have been obtained in the electrostatic floating potential fluctuations of a dc glow discharge plasma. Power spectrum exhibits a power law which is compatible with the requirement for SOC systems. Also the estimated value of the Hurst exponent (self similarity parameter), H being greater than 0.5, along with an algebraic decay of the autocorrelation function, indicate the presence of temporal long-range correlations, as may be expected from SOC dynamics. This type of observations in our opinion has been reported for the first time in a glow discharge system.

Chaos in low-dimensional Lotka–Volterra models of competition

The occurrence of chaos in basic Lotka–Volterra models of four competing species is studied. A brute-force numerical search conditioned on the largest Lyapunov exponent (LE) indicates that chaos occurs in a narrow region of parameter space but is robust to perturbations. The dynamics of the attractor for a maximally chaotic case are studied using symbolic dynamics, and the question of self-organized critical behaviour (scale-invariance) of the solution is considered.

The dynamics of wet granular matter

We present a numerical study of a shear-induced solid–fluid transition in wet granular matter in order to show the self-organized critical behavior close to the transition point. The continuous time simulation is based on a simple model considering both the cohesive forces induced by the adsorbed liquid amount and the repulsive forces due to the hard core interaction of the granules. Dissipation is assumed to be entirely due to the hysteretic character of the cohesive forces. The aim of our analysis concerns the crossover from a solid like behavior to a mobile ergodic state under the influence of an external force field F exceeding the critical force Fc. Diffusion coefficients, dissipation and kinetic order parameters can be expressed as characteristic scaling laws.

Scaling and correlation of atmospheric acid deposition evolutions

Detrended fluctuation analysis (DFA) and power spectrum density are applied to verify the presence of temporal scaling behavior and long-range persistence (LRP) of weekly hydrogen ion deposition (WHD), NO3- and SO42- deposition series in National Atmospheric Deposition Program, USA, for the period 1978–2001. The results show a common scaling behavior for all sites analyzed. Two distinct scaling regions are identified by DFA1, one corresponding to 1 month to 1 year and the other to 1 year to 5 years. The WHD series obey power-law in two temporal regimes respectively with mean DFA1 scaling exponents α1≈0.68 and α2≈0.45, implying the presence of LRP in the acid deposition series and there is a tendency for a large deposition event to be followed by another large event, and vice versa. For DFA2–DFA4, however, the annual crossover, which divides the temporal scale into two regimes, disappears gradually with the order q of DFAq increasing, and the two scaling regimes turn to share the same scaling exponent close to α1. The result indicates that the scaling behavior exits in the two regimes with the same scaling exponent α1, and LRP prevails during the examined 1-month to 5-years scale. NO3- and SO42- deposition evolve the same way as WHD does, implying the pollutants involving in acid deposition may share some prominent mechanism controlling their evolutions. We ascribe the long-range power-law scaling of acid deposition evolutions to the self-organized critical behavior of atmosphere under pollution stress and it should be considered in the trend prediction of acid deposition as an important factor.

Scale-free avalanche dynamics in the stock market

Self-organized criticality (SOC) has been claimed to play an important role in many natural and social systems. In the present work we empirically investigate the relevance of this theory to stock-market dynamics. Avalanches in stock-market indices are identified using a multi-scale wavelet-filtering analysis designed to remove Gaussian noise from the index. Here, new methods are developed to identify the optimal filtering parameters which maximize the noise removal. The filtered time series is reconstructed and compared with the original time series. A statistical analysis of both high-frequency Nasdaq E-mini Futures and daily Dow Jones data is performed. The results of this new analysis confirm earlier results revealing a robust power-law behaviour in the probability distribution function of the sizes, duration and laminar times between avalanches. This power-law behaviour holds the potential to be established as a stylized fact of stock market indices in general. While the memory process, implied by the power-law distribution of the laminar times, is not consistent with classical models for SOC, we note that a power-law distribution of the laminar times cannot be used to rule out self-organized critical behaviour.

Self-Organized Forest-Fires Near the Critical Time

We consider a forest-fire model which, somewhat informally, is described as follows: Each site (vertex) of the square lattice is either vacant or occupied by a tree. Vacant sites become occupied at rate 1. Further, each site is hit by lightning at rate λ. This lightning instantaneously destroys (makes vacant) the occupied cluster of the site. This model is closely related to the Drossel-Schwabl forest-fire model, which has received much attention in the physics literature. The most interesting behaviour seems to occur when the lightning rate goes to zero. In the physics literature it is believed that then the system has so-called self-organized critical behaviour. We let the system start with all sites vacant and study, for positive but small λ, the behaviour near the ‘critical time’ t c , defined by the relation 1 − exp(− t c ) = p c , the critical probability for site percolation. Intuitively one might expect that if, for fixed t > t c , we let simultaneously λ tend to 0 and m to ∞, the probability that some tree at distance smaller than m from O is burnt before time t goes to 1. However, we show that under a percolation-like assumption (which we can not prove but believe to be true) this intuition is false. We compare with the case where the square lattice is replaced by the directed binary tree, and pose some natural open problems.

Self-organized criticality in one-dimensional sandpile model with avalanche probability included

Proposed an one-dimensional sandpile model which include avalanche probability, and performed computer simulation by cellular automata method. The results show that there are two critical points p1 and p2 when avalanche probability p transits from 0 to 1. The self-organized criticality(SOC) behavior can be found in the model when p12. There is a sharp transition between the trivial behaviour and the SOC behaviour in the model. When there is SOC, the SOC behaviour is universal, the two critical exponents are 1.50±0.02 and 1.58±0.15. With the model, the SOC phenomenon appearing in the experiment of one-demensional rice-pile is well explained.

Dynamic critical approach to self-organized criticality

A dynamic scaling ansatz for the approach to the self-organized critical (SOC) regime is proposed and tested by means of extensive simulations applied to the Bak-Sneppen model (BS), which exhibits robust SOC behavior. Considering the short-time scaling behavior of the density of sites [rho(t)] below the critical value, it is shown that (i) starting the dynamics with configurations such that rho(t=0)-->0 one observes an initial increase of the density with exponent theta=0.12(2); (ii) using initial configurations with rho(t=0)-->1, the density decays with exponent delta=0.47(2). It is also shown that the temporal autocorrelation decays with exponent Ca=0.35(2). Using these dynamically determined critical exponents and suitable scaling relationships, all known exponents of the BS model can be obtained, e.g., the dynamical exponent z=2.10(5), the mass dimension exponent D=2.42(5), and the exponent of all returns of the activity tauALL=0.39(2), in excellent agreement with values already accepted and obtained within the SOC regime.

Fibre slippage in cyclically perturbed fibrous assemblies

The dynamics of fibre slippage within general non-bonded fibrous assemblies is studied in the situation where the assembly is subjected to general small cyclic loads. Two models are proposed. The first is applicable when the general cyclic loading is complemented by an occasional tugging force on one end of a fibre, which causes it to gradually withdraw from the assembly, such as might occur during the pilling of a textile. The second considers the situation in which the cyclic perturbations act around a constant background load applied to the assembly. The dynamics is reminiscent of self-organized critical behaviour. This model is applied to predict the progressive elongation of a single yarn during weaving.

Continuous Strain Bursts in Crystalline and Amorphous Metals During Plastic Deformation by Nanoindentation

Using depth-sensing indentation with sub-nanometer displacement resolution, the plastic deformation of a range of materials, including a metallic glass, amorphous selenium, Ni3Al, pure Nb, Al, Cu, and Zn metals, and an Al-Mg alloy, has been investigated at room temperature. In amorphous selenium, even the sub-nanometer displacement resolution of the nanoindentation technique cannot reveal any strain burst during deformation at room temperature. In all other metals studied, what may appear to be smooth load-displacement curves at macroscopic scale during indentation deformation in fact turn out to consist of a continuous series of random bursts of the nanometer scale. The occurrence probability of the bursts is found to decrease at increasing burst size. In all of the crystalline metals and alloys studied, the size distribution of the strain bursts seems to follow an exponential law with a characteristic length scale. The absence of the self-organized critical behavior is likely a result of the small size of the strained volume in the nanoindentation situation, which gives rise to a constraint of a characteristic strain. In the metallic glass sample, due to the limited range of the burst sizes encountered, whether the deformation bursts follow an exponential or a power-law behavior corresponding to self-organized criticality is inconclusive. From a theoretical viewpoint based on the Shannon entropy, the exponential distribution is the most likely distribution at a given mean burst size, and this is thought to be the reason for its occurrence in different materials.

Invasion percolation between two sites

We investigate the process of invasion percolation between two sites (injection and extraction sites) separated by a distance r in two-dimensional lattices of size L. Our results for the nontrapping invasion percolation model indicate that the statistics of the mass of invaded clusters is significantly dependent on the local occupation probability (pressure) Pe at the extraction site. For Pe = 0, we show that the mass distribution of invaded clusters P(M) follows a power-law P(M) approximately M(-alpha) for intermediate values of the mass M, with an exponent alpha = 1.39+/-0.03. When the local pressure is set to Pe = Pc, where Pc corresponds to the site percolation threshold of the lattice topology, the distribution P(M) still displays a scaling region, but with an exponent alpha = 1.02+/-0.03. This last behavior is consistent with previous results for the cluster statistics in standard percolation. In spite of these differences, the results of our simulations indicate that the fractal dimension of the invaded cluster does not depend significantly on the local pressure Pe and it is consistent with the fractal dimension values reported for standard invasion percolation. Finally, we perform extensive numerical simulations to determine the effect of the lattice borders on the statistics of the invaded clusters and also to characterize the self-organized critical behavior of the invasion percolation process.

Scaling properties of air pollution in Athens, Greece and Baltimore, Maryland

Abstract Detrended fluctuation analysis is applied to the Athens air-pollution time-series consisting of hourly observations of ozone, nitrogen oxides, and particulate matter obtained at five air-pollution monitoring stations during 1987–2003. Persistent power-law correlations in the fluctuations of daytime and nighttime ozone concentrations with lag times ranging from 1 week to 5 years were detected with more intense correlations (“stronger memory”) during daytime. The fluctuations of nitrogen oxide concentrations exhibit similar behavior. Finally, persistent power-law correlations from about 4 h to 9 months were found in PM10 fluctuations in Athens. Long-range correlations for lag times from about 4 h to 2 weeks were also detected for PM2.5 fluctuations in a 6-month data set collected at the University of Maryland “Supersite” in East Baltimore. Although the origin of the long-range behaviour of air-pollution merits further investigation, these findings presumably, reflect short and longer term source/meteorological features that are probably related to the self-organized critical behavior of the atmosphere. These findings provide power-law relationships which may substantially contribute to the development of more reliable simulating models for the ozone, nitrogen oxides, and particulate matter fluctuations.

Self-organized criticality induced by quenched disorder: Experiments on flux avalanches inNbHxfilms

We present an experimental study of the influence of quenched disorder on the distribution of flux avalanches in type-II superconductors. In the presence of much quenched disorder, the avalanche sizes are power-law distributed and show finite-size scaling, as expected from self-organized criticality (SOC). Furthermore, the shape of the avalanches is observed to be fractal. In the absence of quenched disorder, a preferred size of avalanches is observed and avalanches are smooth. These observations indicate that a certain minimum amount of disorder is necessary for SOC behavior. We relate these findings to the appearance or nonappearance of SOC in other experimental systems, particularly piles of sand.