Nonconservative earthquake model of self-organized criticality on a random graph

We numerically investigate the Olami-Feder-Christensen model on a quenched random graph. Contrary to the case of annealed random neighbors, we find that the quenched model exhibits self-organized criticality deep within the nonconservative regime. The probability distribution for avalanche size obeys finite size scaling, with universal critical exponents. In addition, a power law relation between the size and the duration of an avalanche exists. We propose that this may represent the correct mean-field limit of the model rather than the annealed random neighbor version.     

Cascades and self-organized criticality

We generalize the model of self-organized critical systems to cases where due to some internal degrees of freedom the local conservation law is violated. This can be realized by taking a transfer ratio different from the critical one in a sand pile model (global violation) or allowing fluctuations around the critical ratio (local violation). In the first case the deviation from the critical ratioR is a critical parameter and the characteristic avalanche size diverges as |R|^– . In the second case the global conservation assures criticality; however, our numerical results indicate that the model is in a new universality class.On leave from Institute for Experimental Physics, JATE University, Dóm tèr 9, Szeged, H-6720 Hungary.On leave from Institute for Technical Physics, H-1325 Budapest, Hungary.     

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.     

Fluctuations in mean-field self-organized criticality

We present two models that exhibit self-organized criticality at the mean-field level. These can be variously interpreted in epidemiological or chemical reaction terms. By studying the master equation for these models we find, however, that only in one of them does the self-organized critical behavior survive in the face of fluctuations. For this model we show the spectrum of the evolution operator to have spectral collapse, i.e., instead of a gap, as would occur in noncritical behavior, there are eigenvalues that approach zero as an inverse power of system size.     

Multiscale self-organized criticality and powerful X-ray flares

A combination of spectral and moments analysis of the continuous X-ray flux data is used to show consistency of statistical properties of the powerful solar flares with 2D BTW prototype model of self-organized criticality.Received: 25 June 2003, Published online: 24 October 2003PACS: 05.65. + b Self-organized systems - 05.45.Tp Time series analysis - 96.60.Rd Flares, bursts, and related phenomena     

Controlling self-organized criticality in complex networks

A control scheme to reduce the size of avalanches of the Bak-Tang-Wiesenfeld model on complex networks is proposed. Three network types are considered: those proposed by Erdős-Renyi, Goh-Kahng-Kim, and a real network representing the main connections of the electrical power grid of the western United States. The control scheme is based on the idea of triggering avalanches in the highest degree nodes that are near to become critical. We show that this strategy works in the sense that the dissipation of mass occurs most locally avoiding larger avalanches. We also compare this strategy with a random strategy where the nodes are chosen randomly. Although the random control has some ability to reduce the probability of large avalanches, its performance is much worse than the one based on the choice of the highest degree nodes. Finally, we argue that the ability of the proposed control scheme is related to its ability to reduce the concentration of mass on the network.     

Power laws and self-organized criticality in theory and nature

Power laws and distributions with heavy tails are common features of many complex systems. Examples are the distribution of earthquake magnitudes, solar flare intensities and the sizes of neuronal avalanches. Previously, researchers surmised that a single general concept may act as an underlying generative mechanism, with the theory of self organized criticality being a weighty contender.     

Analysis of self-organized criticality in weighted coupled systems

A weighted Olami, Feder, and Christensen (OFC) model, improving the redistribution rule of the original model, has been introduced. It can be seen as a generalization of the OFC model and exhibits Self-organized criticality (SOC) behavior, too. The stress evolution process has been accelerated and the nontrivial relationship between the exponent of 1/f and the control parameter has been reported. Although our model is simple, we obtained more reasonable avalanche dimensions than with the previous model.     

Universal phenomenon of self-organized criticality in magnetic flux dynamics

Magnetization curves of square arrays of Josephson junctions of two basic types were investigated: superconductor–insulator–superconductor (SIS) and superconductor–normal metal–superconductor (SNS).

Magnetic flux avalanches were observed in SIS arrays. A statistical analysis of flux avalanches showed that their size distribution can be described by a power law with a crossover where the exponent n varies from −1.2 for small avalanches to −3.5 for the large ones. Such a behavior of avalanches is interpreted as the self-organized criticality (SOC) manifestation. In SNS arrays, the flux avalanches were not observed, but a considerable asymmetry of a hysteresis curve was revealed.     

Number theoretic example of scale-free topology inducing self-organized criticality

In this Letter we present a general mechanism by which simple dynamics running on networks become self-organized critical for scale-free topologies. We illustrate this mechanism with a simple arithmetic model of division between integers, the division model. This is the simplest self-organized critical model advanced so far, and in this sense it may help to elucidate the mechanism of self-organization to criticality. Its simplicity allows analytical tractability, characterizing several scaling relations. Furthermore, its mathematical nature brings about interesting connections between statistical physics and number theoretical concepts. We show how this model can be understood as a self-organized stochastic process embedded on a network, where the onset of criticality is induced by the topology.     

On a forest fire model with supposed self-organized criticality

We study a stochastic forest fire model introduced by P. Baket al. as a model showing self-organized criticality. This model involves a growth parameterp, and the criticality is supposed to show up in the limitp0. By simulating the model on much larger lattices, and with much smaller values ofp, we find that the correlations with longest range do not show a nontrivial critical phenomenon in this limit, though we cannot rule out percolation-like critical behavior on a smaller but still divergent length scale. In contrast, the model shows nontrivialdeterministic evolution over time scales 1/p in the limitp0.     

Carbon nanostructures as an example of the self-organized criticality

The mechanism of self-organized criticality in the region of the nano- and microsizes has been studied experimentally using carbon multiwalled nanotubes and powder polycrystalline graphite as examples. A classical experiment on the formation of samples in the form of “sand heaps” has been described. The specific features of the experiment are the measurement of the electrical resistance of lateral layers at fixed tilt angles of the surface, on which the sample is formed, and the performance of the experiment in the pre-breakdown region of voltages. The observation of the power law for the dependence of the electrical resistance of the samples under study on the number of material portions has demonstrated the manifestation of the mechanism of the self-organized criticality.     

Percolation, self-organized criticality, and electrical instability in carbon nanostructures

The processes of avalanche formation, percolation, and electrical instability have been investigated experimentally using multi-walled and single-walled carbon nanotubes as an example. The performed investigations are based on the comparison of electrical conductivity dynamics in classical experiments, such as the “sand heap,” the two-dimensional grid of resistances with a stochastic node blocking, and the nanosecond percolation in a mode of electrical instability in nanotube tangles/granules. The regularities of mechanisms are revealed and the general concept is formulated.     

Coexistence of self-organized criticality and intermittent turbulence in the solar corona

An extended data set of extreme ultraviolet images of the solar corona provided by the SOHO spacecraft is analyzed using statistical methods common to studies of self-organized criticality (SOC) and intermittent turbulence (IT). The data exhibit simultaneous hallmarks of both regimes: namely, power-law avalanche statistics as well as multiscaling of structure functions for spatial activity. This implies that both SOC and IT may be manifestations of a single complex dynamical process entangling avalanches of magnetic energy dissipation with turbulent particle flows.