Effect of Correlations on the Exponents for the Power-Law Distributions in Self-Organized Criticality

The origin of power-law distributions in self-organized criticality is investigated by treating the variation of the number of active sites in the system as a stochastic process. An avalanche is mapped to a first-return random-walk process in a one-dimensional lattice. In order to understand the reason of variant exponents for the power-law distributions in different self-organized critical systems, we introduce the correlations among evolution steps. Power-law distributions of the lifetime and spatial size are found when the random walk is unbiased with equal probability to move in opposite directions. It is found that the longer the correlation length, the smaller values of the exponents for the power-law distributions.     

On Origin of Power-Law Distributions in Self-Organized Criticality from Random Walk Treatment

The origin of power-law distributions in self-organized criticality is investigated by treating the variation of the number of active sites in the system as a stochastic process. An avalanche is then regarded as a first-return random walk process in a one-dimensional lattice. We assume that the variation of the number of active sites has three possibilities in each update： to increase by 1 with probability f1, to decrease by 1 with probability f2, or remain unchanged with probability 1 - f1 - f2. This mimics the dynamics in the system. Power-law distributions of the lifetime are found when the random walk is unbiased with equal probability to move in opposite directions. This shows that power-law distributions in self-organized criticality may be caused by the balance of competitive interactions.     

Flat-head power-law, size-independent clustering, and scaling of coevolutionary scale-free networks

Scale-free topology and high clustering coexist in some real networks, and keep invariant for growing sizes of the systems. Previous models could hardly give out size-independent clustering with selforganized mechanism when succeeded in producing power-law degree distributions. Always ignored, some empirical statistic results display flat-head power-law behaviors. We modify our recent coevolutionary model to explain such phenomena with the inert property of nodes to retain small portion of unfavorable links in self-organized rewiring process. Flat-head power-law and size-independent clustering are induced as the new characteristics by this modification. In addition, a new scaling relation is found as the result of interplay between node state growth and adaptive variation of connections.     

Avalanche dynamics of idealized neuron function in the brain on an uncorrelated random scale-free network

We study a simple model for a neuron function in a collective brain system. The neural network is composed of an uncorrelated configuration model (UCM) for eliminating the degree correlation of dynamical processes. The interaction of neurons is assumed to be isotropic and idealized. These neuron dynamics are similar to biological evolution in extremal dynamics with locally isotropic interaction but has a different time scale. The functioning of neurons takes place as punctuated patterns based on avalanche dynamics. In our model, the avalanche dynamics of neurons exhibit self-organized criticality which shows power-law behavior of the avalanche sizes. For a given network, the avalanche dynamic behavior is not changed with different degree exponents of networks, γ≥2.4 and various refractory periods referred to the memory effect, T_r. Furthermore, the avalanche size distributions exhibit power-law behavior in a single scaling region in contrast to other networks. However, return time distributions displaying spatiotemporal complexity have three characteristic time scaling regimes Thus, we find that UCM may be inefficient for holding a memory.     

Cluster growth poised on the edge of break-up: Size distributions with small exponent power-laws

In many naturally occurring growth processes, cluster size distributions of power-law form n(s)∝s^−τ with small exponents 0<τ<1 are observed. We suggest here that such distributions emerge naturally from cluster growth, where size dependent aggregation is counterbalanced by size dependent break-up. The model used in the study is a simple reaction kinetic model including only monomer-cluster processes. It is shown that under such conditions power-law size distributions with small exponents are obtained. Therefore, the results suggest that the ubiquity of small exponent power-law distributions is related to the growth process, where aggregation driven cluster growth is poised on the edge of cluster break-up.     

Delayed self-organized criticality and earthquake modeling

Delayed self-organized criticality is defined. It is shown to preserve the power-law behavior of self-organized criticality with a significant change in the exponents. A delayed version of the Ito—Matsuzaki model for earthquakes is constructed and studied. This model explains some fractal features of earthquakes as well as the Gutenberg-Richter and Omori laws. Furthermore theb value obtained from the delayed model is closer to observations than theb value of the undelayed model.     

Critical Behaviors in a Stochastic Local Limited One-Dimensional Rice-Pile Model

A stochastic local limited one-dimensional rice-pile model is numerically investigated. The distributions for avalanche sizes have a clear power-law behavior and it displays a simple finite size scaling. We obtain the avalanche exponents τ_s=1.54±0.10, β_s=2.17±0.10 and τ_T=1.80±0.10, β_T=1.46±0.10. This self-organized critical model belongs to the same universality class with the Oslo rice-pile model studied by K. Christensen et al. [Phys. Rev. Lett. 77 (1996) 107], a rice-pile model studied by L.A.N. Amaral et al. [Phys. Rev. E 54 (1996) 4512], and a simple deterministic self-organized critical model studied by M.S. Vieira [Phys. Rev. E 61 (2000) 6056].     

Formation of power-law size distributions of defects during fracture of materials

The experimental data on the surface relief of loaded ribbons of an amorphous alloy have been obtained. The distributions of surface defects formed under loading have been analyzed using the wavelet transform and box counting method. Moreover, the data on the time accumulation of microcracks in the volume of a loaded granite specimen have been examined. It has been shown that power-law size distributions of defects (scaling) appear on the surface and in the bulk before fracture. It has been revealed that the appearance of the power-law distributions is one of the indications of the formation of the self-organized critical state. The formation of the self-organized critical state in the bulk and on the surface of the material has been considered. It has been established that the formation of the self-organized critical state precedes the fracture of a solid.     

Environment-dependent blinking of single semiconductor nanocrystals and statistical aging of ensembles

We compare decreasing fluorescence signals from ensembles with the blinking statistics of individual nanocrystals in various environments. For most substrates, the ensemble decay follows a power-law of time, the exponent being the difference of the power-law exponents of on- and off-time distributions. The decay exponent is also found to depend on substrate. We discuss possible mechanisms for this dependence, in conjunction with previously published models.     

Temporal generation of power-law distributions: A universal ‘oligarchy mechanism’

We present a universal mechanism for the temporal generation of power-law distributions with arbitrary integer-valued exponents.     

The critical properties of the agent-based model with environmental-economic interactions

The steady-state and nonequilibrium properties of the model of environmental-economic interactions are studied. The interacting heterogeneous agents are simulated on the platform of the emission dynamics of cellular automaton. The diffusive emissions are produced by the factory agents, and the local pollution is monitored by the randomly walking (mobile) sensors. When the threshold concentration is exceeded, a feedback signal is transmitted from the sensor to the nearest factory that affects its actual production rate. The model predicts the discontinuous phase transition between safe and catastrophic ecology. Right at the critical line, the broad-scale power-law distributions of emission rates have been identified. The power-law fluctuations are triggered by the screening effect of factories and by the time delay between the environment contamination and its detection. The system shows the typical signs of the self-organized critical systems, such as power-law distributions and scaling laws. The text was submitted by the authors in English.     

Return probability for random walks on scale-free complex trees

The time course of random processes usually differs depending on the topology of complex networks which are a substrate for the process. However, as this Letter demonstrates, the first-return as well as the survival probabilities for random walks on the scale-free (SF) trees decay in time according to the same invariant power-law behavior. This means that both quantities are independent of the node power-law degree distributions which are distinguished by different scaling exponents. It is also shown here that the crucial property of the networks, affecting the dynamics of random walks, is their tree-like topology and not SF architecture. All analytical results quantifying these predictions have been verified through extensive computer simulations.     

Precursory phenomena associated with large avalanches in the long-range connective sandpile (LRCS) model

Reduction in b-values before a large earthquake is a very popular topic for discussion. This study proposes an alternative sandpile model being able to demonstrate reduction in scaling exponents before large events through adaptable long-range connections. The distant connection between two separated cells was introduced in the sandpile model. We found that our modified long-range connective sandpile (LRCS) system repeatedly approaches and retreats from a critical state. When a large avalanche occurs in the LRCS model, accumulated energy dramatically dissipates and the system simultaneously retreats from criticality. The system quickly approaches the critical state accompanied by the increase in the slopes of the power-law frequency-size distributions of events. Afterwards, and most interestingly, the power-law slope declines before the next large event. The precursory b-value reduction before large earthquakes observed from earthquake catalogues closely mimics the evolution in power-law slopes for the frequency-size distributions of events derived in the LRCS models. Our paper, thus, provides a new explanation for declined b-values before large earthquakes.     

Weighted scale-free networks with variable power-law exponents

We present a weighted scale-free network model, in which the power-law exponents can be controlled by the model parameters. The network is generated through the weight-driven preferential attachment of new nodes to existing nodes and the growth of the weights of existing links. The simplicity of the model enables us to derive analytically the various statistical properties, such as the distributions of degree, strength, and weight, the degree-strength and degree-weight relationship, and the dependencies of these power-law exponents on the model parameters. Finally, we demonstrate that networks of words, coauthorship of researchers, and collaboration of actor/actresses are quantitatively well described by this model.     

Black swans,power laws,and dragon-kings: Earthquakes,volcanic eruptions,landslides, wildfires,floods, and SOC models

Extreme events that change global society have been characterized as black swans. The frequency-size distributions of many natural phenomena are often well approximated by power-law (fractal) distributions. An important question is whether the probability of extreme events can be estimated by extrapolating the power-law distributions. Events that exceed these extrapolations have been characterized as dragon-kings. In this paper we consider extreme events for earthquakes, volcanic eruptions, wildfires, landslides and floods. We also consider the extreme event behavior of three models that exhibit self-organized criticality (SOC): the slider-block, forest-fire, and sand-pile models. Since extrapolations using power-laws are widely used in probabilistic hazard assessment, the occurrence of dragon-king events have important practical implications.     

Self-organized Criticality in a Model Based on Neural Network

Based on the LISSOM neural network model, we introduce a model to investigate self-organized criticality in the activity of neural populations. The influence of connection (synapse) between neurons has been adequately considered in this model. It is found to exhibit self-organized criticality (SOC) behavior under appropriate conditions. We also find that the learning process has promotive influence on emergence of SOC behavior. In addition, we analyze the influence of various factors of the model on the SOC behavior, which is characterized by the power-law behavior of the avalanche size distribution.     

Continuum percolation conductivity exponents in restricted domains

Conductivity behavior of continuum percolation in restricted two-dimensional domains is simulated by considering systems of randomly distributed disks. The domain is restricted in that conducting objects are permitted to lie in only a portion of the domain. Such a restricted domain might better approximate some natural systems. Simulations of two-dimensional systems, based on three distributions of local conductances, are examined and found to demonstrate a power-law behavior with conductivity exponents smaller than those arising in regular lattice and continuum percolation     

Degree distributions of growing networks

The in-degree and out-degree distributions of a growing network model are determined. The in-degree is the number of incoming links to a given node (and vice versa for out-degree). The network is built by (i) creation of new nodes which each immediately attach to a preexisting node, and (ii) creation of new links between preexisting nodes. This process naturally generates correlated in-degree and out-degree distributions. When the node and link creation rates are linear functions of node degree, these distributions exhibit distinct power-law forms. By tuning the parameters in these rates to reasonable values, exponents which agree with those of the web graph are obtained.     

Self-organization of structures and networks from merging and small-scale fluctuations

We discuss merging-and-creation as a self-organizing process for scale-free topologies in networks. Three power-law classes characterized by the power-law exponents , 2 and are identified and the process is generalized to networks. In the network context the merging can be viewed as a consequence of optimization related to more efficient signaling.     

Study on probability distributions for evolution in modified extremal optimization

It is widely believed that the power-law is a proper probability distribution being effectively applied for evolution in τ-EO (extremal optimization), a general-purpose stochastic local-search approach inspired by self-organized criticality, and its applications in some NP-hard problems, e.g., graph partitioning, graph coloring, spin glass, etc. In this study, we discover that the exponential distributions or hybrid ones (e.g., power-laws with exponential cutoff) being popularly used in the research of network sciences may replace the original power-laws in a modified τ-EO method called self-organized algorithm (SOA), and provide better performances than other statistical physics oriented methods, such as simulated annealing, τ-EO and SOA etc., from the experimental results on random Euclidean traveling salesman problems (TSP) and non-uniform instances. From the perspective of optimization, our results appear to demonstrate that the power-law is not the only proper probability distribution for evolution in EO-similar methods at least for TSP, the exponential and hybrid distributions may be other choices.