Self-organized Critical Model Research Articles

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 model possess the discontinuous transition between the safe and catastrophic ecology. Right at the critical line, the broad-scale power-law distributions of emission rates have been identified. Their relationship to Zipf's law and models of self-organized criticality is discussed.

From Solar and Stellar Flares to Coronal Heating: Theory and Observations of How Magnetic Reconnection Regulates Coronal Conditions

There is currently no explanation of why the corona has the temperature and density it has. We present a model which explains how the dynamics of magnetic reconnection regulates the conditions in the corona. A bifurcation in magnetic reconnection at a critical state enforces an upper bound on the coronal temperature for a given density. We present observational evidence from 107 flares in 37 sun-like stars that stellar coronae are near this critical state. The model may be important to self-organized criticality models of the solar corona.

Avalanches in networks of weakly coupled phase-shifting rotators

The networks of phase-shifting rotators interacting through exchange of weak ±-kicks are considered. Such a rotator consists of a particle rotating on a circle which at some discrete mo- ments receives some ±-kicks. We assume that the kicks are not of mechanical character: they change a particle's position but not the rate. A comparison of the rotator networks with the BTW model of self-organized criticality, Burridge-Knopoff's model in seismicity, Herz-Hopfield neural networks, and the Turing-Smale system in biological cells is presented. This work studies the avalanches in rotator networks — the events when a number of rotators almost simultaneously hit some threshold levels. The asymptotic relations linking distribution of avalanches with the architecture of a network are proved. The equivalence of two well-known power-law conjectures, in lattice models of statistical physics and in interacting threshold systems, is established.

Predicting Solar Flares by Data Assimilation in Avalanche Models

Data assimilation techniques, developed in the past two decades mainly for weather prediction, produce better forecasts by taking advantage of both theoretical/numerical models and real-time observations. In this paper, we explore the possibility of applying the four-dimensional variational data assimilation (4D-VAR) technique to the prediction of solar flares. We do so in the context of a continuous version of the classical cellular-automaton-based self-organized critical avalanche models of solar flares introduced by Lu and Hamilton (Astrophys. J. 380, L89, 1991). Such models, although a priori far removed from the physics of magnetic reconnection and magnetohydrodynamical evolution of coronal structures, nonetheless reproduce quite well the observed statistical distribution of flare characteristics. We report here on a large set of data assimilation runs on synthetic energy release time series. Our results indicate that, despite the unpredictable (and unobservable) stochastic nature of the driving/triggering mechanism within the avalanche model, 4D-VAR succeeds in producing optimal initial conditions that reproduce adequately the time series of energy released by avalanches and flares. This is an essential first step toward forecasting real flares.

On applicability of the RTL prognostic algorithms and energy estimation to Sakhalin seismicity

Premonitory phases (seismic quiescence and foreshock activity) have been retrospectively identified before the Neftegorsk and Uglegorsk earthquakes using the RTL technique. The probabilities that these phases were accidental are less than 1 and 2%, respectively. This allows an optimistic assessment of the possibility of applying this technique to seismicity at Sakhalin. The estimates of the time and energy class for the two earthquakes, using a model of self-organized seismic criticality, proved to be unconvincing because obvious acceleration of the seismic process prior to these seismic events did not occur. The applicability of this approach to the seismicity at Sakhalin should be tested for future large earthquakes. The regional Sakhalin catalog for 1980–2000, with a lowest completely reported energy class of K = 8 (lent by the Geophysical Service, Russian Academy of Sciences) was used as the database for this study.

Self-organized critical models of earthquakes

In this paper we briefly review few self-organized critical (SOC) models of the phenomenon of earthquakes. For example, the two-dimensional non-conservative SOC model of Olami, Feder and Christensen (OFC) has been described. It is known that the effect of the fixed boundary on this model is very strong. It has been recently observed that imposition of a moving boundary condition helps to remove the strong non-uniformity originated from the fixed boundary. A generalized spatio-temporal scaling for the recurrence time distribution was proposed by Bak et al. which was later confirmed by Corral. We studied the same scalings on the conservative OFC model with moving boundary condition.

Spatial scaling of optical fluctuations during substorm-time aurora

Abstract. A study of statistical features of auroras during substorm activity is presented, emphasizing characteristics which are commonly applied to turbulent flows. Data from all-sky television (TV) observations from the Barentsburg observatory (Svalbard) have been used. Features of the probability density function (PDF) of auroral fluctuations have been examined at different spatial scales. We find that the observed PDFs generally have a non-Gaussian, heavy-tailed shape. The generalized structure function (GSF) for the auroral luminosity fluctuations has been analyzed to determine the scaling properties of the higher (up to 6) order moments, and the evolution of the scaling indices during the actual substorm event has been determined. The scaling features obtained can be interpreted as signatures of turbulent motion of the magnetosphere-ionosphere plasma. Relations to previously obtained results of avalanche analysis of the same event, as well as possible implications for the validity of self-organized criticality models and turbulence models of the substorm activity, are discussed.

Self-organized Criticality in a Modified EvolutionModel on Generalized Barabási–Albert Scale-Free Networks

A modified evolution model of self-organized criticality ongeneralized Barabási–Albert (GBA) scale-free networks isinvestigated. In our model, we find that spatial and temporalcorrelations exhibit critical behaviors. More importantly, thesecritical behaviors change with the parameter b, which weightsthe distance in comparison with the degree in the GBA networkevolution.

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 model possesses the discontinuous transition between the safe and catastrophic ecology. Right at the critical line, the broad-scale power-law distributions of emission rates have been identified. Their relationship to Zipf's law and models of self-organized criticality is discussed.

Self-organized critical earthquake model with moving boundary

A globally driven self-organized critical model of earthquakes with conservative dynamics has been studied. An open but moving boundary condition has been used so that the origin (epicenter) of every avalanche (earthquake) is at the center of the boundary. As a result, all avalanches grow in equivalent conditions and the avalanche size distribution obeys finite size scaling excellent. Though the recurrence time distribution of the time series of avalanche sizes obeys well both the scaling forms recently observed in analysis of the real data of earthquakes, it is found that the scaling function decays only exponentially in contrast to a generalized gamma distribution observed in the real data analysis. The non-conservative version of the model shows periodicity even with open boundary.

The Ising model in a Bak–Tang–Wiesenfeld sandpile

We study the spin-1 Ising model with non-local constraints imposed by the Bak–Tang–Wiesenfeld sandpile model of self-organized criticality (SOC). The model is constructed as if the sandpile is being built on a (honeycomb) lattice with Ising interactions. In this way we combine two models that exhibit power-law decay of correlation functions characterized by different exponents. We discuss the model properties through an order parameter and the mean energy per node, as well as the temperature dependence of their fourth-order Binder cumulants. We find: (i) a thermodynamic phase transition at a finite T c between paramagnetic and antiferromagnetic phases, and (ii) that above T c the correlation functions decay in a way typical of SOC. The usual thermodynamic criticality of the two-dimensional Ising model is not affected by SOC constraints (the specific heat critical exponent α ≈ 0 ), nor are SOC-induced correlations affected by the interactions of the Ising model. Even though the constraints imposed by the SOC model induce long-range correlations, as if at standard (thermodynamic) criticality, these SOC-induced correlations have no impact on the thermodynamic functions.

Convergence dynamics of the Bak–Sneppen model: Activity rate and waiting time distribution

In this work, we study the convergence dynamics of two independent random configurations of the Bak–Sneppen model of self-organized criticality evolving under the same external noise. A recently proposed measure of the Hamming distance which considers the minimum difference between displaced configurations is used. The displacement evolves in time intermittently. We compute the jump activity rate and waiting time distribution and report on their asymptotic power-law scaling which characterizes the slow relaxation and the absence of typical length and time scales typical of critical dynamical systems.

Regulating economic systems in a multi-trait model of self-organized criticality

We study the effect of size-based regulation on an economic system. A multiple-trait model of self-organized criticality is used to simulate the economic system. The major difference of this work from previous studies is that firm's fitness is not characterized by a single number but by M traits. Each trait represents one aspect of the competitiveness of the firm, and firm's size is one of these traits. Major finding drawn from the present study is that the effectiveness of regulations decreases with increasing M, i.e., size-based regulations are less effective when the overall fitness of a firm is determined by more factors.

Bursty Behaviour of Turbulent Particle and Energy Fluxes in Edge Plasma of HT-7 Tokamak

High time resolution measurements of the electrostatic fluctuations and the turbulent particle and energy fluxes have been performed with a Langmuir probe array in the edge plasma in HT-7 tokamak. Bursty behaviour was observed in the time resolved turbulent fluxes with positive skewness and large kurtosis. The contribution of the large sporadic bursts to the transport losses were estimated. The analysis shows that the turbulent fluxes have different behaviour in different frequency domains and the probability distribution functions (PDFs) of the particle and energy fluxes present two distinct scaling ranges. All these are essentially consistent with the predictions of the self-organized criticality (SOC) model, though further studies are needed.

Theoretical studies of self-organized criticality

These notes are intended to provide a pedagogical introduction to the abelian sandpile model of self-organized criticality, and its related models. The abelian group, the algebra of particle addition operators, the burning test for recurrent states, equivalence to the spanning trees problem are described. The exact solution of the directed version of the model in any dimension is explained. The model's equivalence to Scheidegger's model of river basins, Takayasu's aggregation model and the voter model is discussed. For the undirected case, the solution for one-dimensional lattices and the Bethe lattice is briefly described. Known results about the two dimensional case are summarized. Generalization to the abelian distributed processors model is discussed. Time-dependent properties and the universality of critical behavior in sandpiles are briefly discussed. I conclude by listing some still-unsolved problems.

Critical finite‐size scaling of energy and lifetime probability distributions of auroral emissions

Based on statistical study of approximately 15,500 ultraviolet images of auroral emission regions provided by the UVI experiment on the POLAR spacecraft, we show that energy and duration probability distributions of particle precipitation events obey finite‐size scaling relations indicative of a self‐organized critical (SOC) dynamical state. The revealed relations are invariant with respect to significant changes in the spatial scale of the emission areas, and involve a set of mutually consistent critical exponents providing a quantitative basis for future theoretical studies of multiscale magnetospheric fluctuations. The reported statistical results highlight the importance of cross‐scale coupling in the development of nighttime geomagnetic disturbances and suggest that various manifestations of substorm activity associated with localized magnetic reconnections in the magnetotail (small to large scale substorms, pseudo‐breakups, BBFs and other types of short‐term localized excitations) can be coordinated on the global scale by universal dynamical principle represented by scale‐free avalanching in numerical SOC models.

Dynamics and Entropy in the Zhang Model of Self-Organized Criticality

We give a detailed study of dynamical properties of the Zhang model, including evaluation of topological entropy and estimates for the Lyapunov exponents and the dimension of the attractor. In the thermodynamic limit the entropy goes to zero and the Lyapunov spectrum collapses.

Self-Organized Criticality Model for Brain Plasticity

Networks of living neurons exhibit an avalanche mode of activity, experimentally found in organotypic cultures. Here we present a model that is based on self-organized criticality and takes into account brain plasticity, which is able to reproduce the spectrum of electroencephalograms (EEG). The model consists of an electrical network with threshold firing and activity-dependent synapse strengths. The system exhibits an avalanche activity in a power-law distribution. The analysis of the power spectra of the electrical signal reproduces very robustly the power-law behavior with the exponent 0.8, experimentally measured in EEG spectra. The same value of the exponent is found on small-world lattices and for leaky neurons, indicating that universality holds for a wide class of brain models.

Uniqueness of multi-dimensional infinite volume self-organized critical forest-fire models

In a forest-fire model, each site of the square lattice is either vacant or occupied by a tree. Vacant sites get occupied according to independent rate 1 Poisson processes. Independently at each site ignition occurs according to independent rate lambda Poisson processes. When a site is hit by ignition, then its whole occupied cluster becomes vacant instantaneously. The article studies whether a multi-dimensional infinite volume forest-fire process with given parameter is unique. Under an assumption on the decay of the cluster size distribution, a process that dominates the forest-fire process is used to show uniqueness. If lambda is big enough, then subcritical site percolation shows the correctness of the assumption

Existence of multi-dimensional infinite volume self-organized critical forest-fire models

Consider the following forest-fire model where the possible locations of trees are the sites of a cubic lattice. Each site has two possible states: 'vacant' or 'occupied'. Vacant sites become occupied according to independent rate 1 Poisson processes. Independently, at each site ignition (by lightning) occurs according to independent rate lambda Poisson processes. When a site is ignited, its occupied cluster becomes vacant instantaneously. If the lattice is one-dimensional or finite, then with probability one, at each time the state of a given site only depends on finitely many Poisson events; a process with the above description can be constructed in a standard way. If the lattice is infinite and multi-dimensional, in principle, the state of a given site can be influenced by infinitely many Poisson events in finite time.