Self-organized Critical Model Research Articles

On Graph Parameters Guaranteeing Fast Sandpile Diffusion

The Abelian Sandpile Model (Dhar 1990) is a discrete diffusion process, defined on graphs, which serves as the standard model of self-organized criticality. One is allowed to add sand particles on the nodes of the graph such that each node can stably hold at most some bounded number of particles. The particles flow through the graph as a consequence of surpassing the node capacities, until they reach a special sink node possessing infinite capacity. These simple dynamics give rise to a very interesting Markovian system. The transience class of a sandpile is defined as the maximum number of particles that can be added without making the system recurrent. We identify a small set of key graph properties that guarantee polynomial bounds on transience classes of the sandpile families satisfying them. These properties governing the speed of sandpile diffusion process are volume growth parameters, boundary regularity type properties and non-empty interior type constraints.This generalizes a previous result by Babai and Gorodezky (2007), in which they establish polynomial bounds on the $n\times n$ grid. Indeed the properties we show are based on ideas extracted from their proof as well as the continuous analogs in the theory of harmonic functions.

Singular-Degenerate Multivalued Stochastic Fast Diffusion Equations

We consider singular-degenerate, multivalued stochastic fast diffusion equations with multiplicative Lipschitz continuous noise. In particular, this includes the stochastic sign fast diffusion equation arising from the Bak--Tang--Wiesenfeld model for self-organized criticality. A well-posedness framework based on stochastic variational inequalities (SVI) is developed, characterizing solutions to the stochastic sign fast diffusion equation, previously obtained in a limiting sense only. Aside from generalizing the SVI approach to stochastic fast diffusion equations we develop a new proof of well-posedness, applicable to general diffusion coefficients. In case of linear multiplicative noise, we prove the existence of (generalized) strong solutions, which entails higher regularity properties of solutions than previously known.

A lower bound on the relative entropy with respect to a symmetric probability

Let $\rho$ and $\mu$ be two probability measures on $\mathbb{R}$ which are not the Dirac mass at $0$. We denote by $H(\mu|\rho)$ the relative entropy of $\mu$ with respect to $\rho$. We prove that, if $\rho$ is symmetric and $\mu$ has a finite first moment, then \[ H(\mu|\rho)\geq \frac{\displaystyle{\left(\int_{\mathbb{R}}z\,d\mu(z)\right)^2}}{\displaystyle{2\int_{\mathbb{R}}z^2\,d\mu(z)}}\,,\] with equality if and only if $\mu=\rho$. We give an applicaion to the Curie-Weiss model of self-organized criticality.

UNIVERSAL BEHAVIOR OF X-RAY FLARES FROM BLACK HOLE SYSTEMS

X-ray flares have been discovered in black hole systems, such as gamma-ray bursts, the tidal disruption event Swift J1644+57, the supermassive black hole Sagittarius A$^*$ at the center of our Galaxy, and some active galactic nuclei. Their occurrences are always companied by relativistic jets. However, it is still unknown whether there is a physical analogy among such X-ray flares produced in black hole systems spanning nine orders of magnitude in mass. Here we report the observed data of X-ray flares, and show that they have three statistical properties similar to solar flares, including power-law distributions of energies, durations, and waiting times, which both can be explained by a fractal-diffusive self-organized criticality model. These statistical similarities, together with the fact that solar flares are triggered by a magnetic reconnection process, suggest that all of the X-ray flares are consistent with magnetic reconnection events, implying that their concomitant relativistic jets may be magnetically dominated.

Finite Time Extinction for Stochastic Sign Fast Diffusion and Self-Organized Criticality

We prove finite time extinction for stochastic sign fast diffusion equations driven by linear multiplicative space-time noise, corresponding to the Bak-Tang-Wiesenfeld model for self-organized criticality. This solves a problem posed and left open in several works: [Barbu, Da Prato, R\"ockner, CMP, 2009], [Barbu, Da Prato, R\"ockner, CRMAS, 2009], [Barbu, R\"ockner, CMP, 2012], [Barbu, Da Prato, R\"ockner, JMAA, 2012], [R\"ockner, Wang, JMLS, 2013], [Barbu, MMAS, 2013]. The highly singular-degenerate nature of the drift in interplay with the stochastic perturbation causes the need for new methods in the analysis of mass diffusion and several new estimates and techniques are introduced.

GLOBAL ENERGETICS OF SOLAR FLARES. I. MAGNETIC ENERGIES

We present the first part of a project on the global energetics of solar flares and coronal mass ejections (CMEs) that includes about 400 M- and X-class flares observed with AIA and HMI onboard SDO. We calculate the potential energy, free energy, and the flare-dissipated magnetic energy. We calculate these magnetic parameters using two different NLFFF codes: The COR-NLFFF code uses the line-of-sight magnetic field component $B_z$ from HMI to define the potential field, and the 2D coordinates of automatically detected coronal loops in 6 coronal wavelengths from AIA to measure the helical twist of coronal loops caused by vertical currents, while the PHOT-NLFFF code extrapolates the photospheric 3D vector fields. We find agreement between the two codes in the measurement of free energies and dissipated energies within a factor of $ \approx 3$. The size distributions of magnetic parameters exhibit powerlaw slopes that are approximately consistent with the fractal-diffusive self-organized criticality model. The magnetic parameters exhibit scaling laws for the nonpotential energy, $E_{np} \propto E_p^{1.02}$, for the free energy, $E_{free} \propto E_p^{1.7}$ and $E_{free} \propto B_{\varphi}^{1.0} L^{1.5}$, for the dissipated energy, $E_{diss} \propto E_p^{1.6}$ and $E_{diss} \propto E_{free}^{0.9}$, and the energy dissipation volume, $V \propto E_{diss}^{1.2}$. The potential energies vary in the range of $E_p = 1 \times 10^{31} - 4 \times 10^{33}$ erg, while the free energy has a ratio of $E_{free}/E_p \approx 1%-25%$. The Poynting flux amounts to $F_{flare} \approx 5 \times 10^{8} - 10^{10}$ erg cm$^{-2}$ s$^{-1}$ during flares, which averages to $F_{AR} \approx 6 \times 10^6$ erg cm$^{-2}$ s$^{-1}$ during the entire observation period and is comparable with the coronal heating rate requirement in active regions.

The Abelian distribution

We define the Abelian distribution and study its basic properties. Abelian distributions arise in the context of neural modeling and describe the size of neural avalanches in fully-connected integrate-and-fire models of self-organized criticality in neural systems.

Spatial planning via extremal optimization enhanced by cell-based local search

A new treatment is presented for land use planning problems by means of extremal optimization in conjunction to cell-based neighborhood local search. Extremal optimization, inspired by self-organized critical models of evolution has been applied mainly to the solution of classical combinatorial optimization problems. Cell-based local search has been employed by the author elsewhere in problems of spatial resource allocation in combination with genetic algorithms and simulated annealing. In this paper it complements extremal optimization in order to enhance its capacity for a spatial optimization problem. The hybrid method thus formed is compared to methods of the literature on a specific characteristic problem. It yields better results both in terms of objective function values and in terms of compactness. The latter is an important quantity for spatial planning. The present treatment yields significant compactness values as emergent results.

Deterministically Driven Avalanche Models of Solar Flares

We develop and discuss the properties of a new class of lattice-based avalanche models of solar flares. These models are readily amenable to a relatively unambiguous physical interpretation in terms of slow twisting of a coronal loop. They share similarities with other avalanche models, such as the classical stick–slip self-organized critical model of earthquakes, in that they are driven globally by a fully deterministic energy-loading process. The model design leads to a systematic deficit of small-scale avalanches. In some portions of model space, mid-size and large avalanching behavior is scale-free, being characterized by event size distributions that have the form of power-laws with index values, which, in some parameter regimes, compare favorably to those inferred from solar EUV and X-ray flare data. For models using conservative or near-conservative redistribution rules, a population of large, quasiperiodic avalanches can also appear. Although without direct counterparts in the observational global statistics of flare energy release, this latter behavior may be relevant to recurrent flaring in individual coronal loops. This class of models could provide a basis for the prediction of large solar flares.

Bottom-up model of self-organized criticality on networks

The Bak-Tang-Wiesenfeld (BTW) sandpile process is an archetypal, stylized model of complex systems with a critical point as an attractor of their dynamics. This phenomenon, called self-organized criticality, appears to occur ubiquitously in both nature and technology. Initially introduced on the two-dimensional lattice, the BTW process has been studied on network structures with great analytical successes in the estimation of macroscopic quantities, such as the exponents of asymptotically power-law distributions. In this article, we take a microscopic perspective and study the inner workings of the process through both numerical and rigorous analysis. Our simulations reveal fundamental flaws in the assumptions of past phenomenological models, the same models that allowed accurate macroscopic predictions; we mathematically justify why universality may explain these past successes. Next, starting from scratch, we obtain microscopic understanding that enables mechanistic models; such models can, for example, distinguish a cascade's area from its size. In the special case of a 3-regular network, we use self-consistency arguments to obtain a zero-parameter mechanistic (bottom-up) approximation that reproduces nontrivial correlations observed in simulations and that allows the study of the BTW process on networks in regimes otherwise prohibitively costly to investigate. We then generalize some of these results to configuration model networks and explain how one could continue the generalization. The numerous tools and methods presented herein are known to enable studying the effects of controlling the BTW process and other self-organizing systems. More broadly, our use of multitype branching processes to capture information bouncing back and forth in a network could inspire analogous models of systems in which consequences spread in a bidirectional fashion.

The sandpile scheduler

This paper studies a self-organized criticality model called sandpile for dynamically load-balancing tasks arriving in the form of Bag-of-Tasks in large-scale decentralized system. The sandpile is designed as a decentralized agent system characterizing a cellular automaton, which works in a critical state at the edge of chaos. Depending on the state of the cellular automaton, different responses may occur when a new task is assigned to a resource: it may change nothing or generate avalanches that reconfigure the state of the system. The abundance of such avalanches is in power-law relation with their sizes, a scale-invariant behavior that emerges without requiring tuning or control parameters. That means that large—catastrophic—avalanches are very rare but small ones occur very often. Such emergent pattern can be efficiently adapted for non-clairvoyant scheduling, where tasks are load balanced in computing resources trying to maximize the performance but without assuming any knowledge on the tasks features. The algorithm design is experimentally validated showing that the sandpile is able to find near-optimal schedules by reacting differently to different conditions of workloads and architectures.

MULTI-WAVELENGTH OBSERVATIONS OF THE SPATIO-TEMPORAL EVOLUTION OF SOLAR FLARES WITH AIA/SDO. II. HYDRODYNAMIC SCALING LAWS AND THERMAL ENERGIES

In this study we measure physical parameters of the same set of 155 M and X-class solar flares observed with AIA/SDO as analyzed in Paper I, by performing a {\sl differential emission measure (DEM)} analysis to determine the flare peak emission measure $EM_p$, peak temperature $T_p$, electron density $n_p$, and thermal energy $E_{th}$, in addition to the spatial scales $L$, areas $A$, and volumes $V$ measured in Paper I. The parameter ranges for M and X-class flares are: $\log(EM_p)=47.0-50.5$, $T_p=5.0-17.8$ MK, $n_p=4 \times 10^9-9 \times 10^{11}$ cm$^{-3}$, and thermal energies of $E_{th}=1.6 \times 10^{28}-1.1 \times 10^{32}$ erg. We find that these parameters obey the Rosner-Tucker-Vaiana (RTV) scaling law $T_p^2 \propto n_p L$ and $H \propto T^{7/2} L^{-2}$ during the peak time $t_p$ of the flare density $n_p$, when energy balance between the heating rate $H$ and the conductive and radiative loss rates is achieved for a short instant, and thus enables the applicability of the RTV scaling law. The application of the RTV scaling law predicts powerlaw distributions for all physical parameters, which we demonstrate with numerical Monte-Carlo simulations as well as with analytical calculations. A consequence of the RTV law is also that we can retrieve the size distribution of heating rates, for which we find $N(H) \propto H^{-1.8}$, which is consistent with the magnetic flux distribution $N(\Phi) \propto \Phi^{-1.85}$ observed by Parnell et al.(2009) and the heating flux scaling law $F_H \propto H L \propto B/L$ of Schrijver et al.(2004). The fractal-diffusive self-organized criticality model in conjunction with the RTV scaling law reproduces the observed powerlaw distributions and their slopes for all geometrical and physical parameters and can be used to predict the size distributions for other flare datasets, instruments, and detection algorithms.

Variational Solutions to Nonlinear Diffusion Equations with Singular Diffusivity

We provide existence results for nonlinear diffusion equations with multivalued time-dependent nonlinearities related to convex continuous not coercive potentials. The results in this paper, following a variational principle, state that a generalized solution of the nonlinear equation can be retrieved as a solution of an appropriate minimization problem for a convex functional involving the potential and its conjugate. In the not coercive case, this assertion is conditioned by the validity of a relation between the solution and the nonlinearity. A sufficient condition, under which this relation is true, is given. At the end, we present a discussion on the solution existence for a particular equation describing a self-organized criticality model.

STEREO/Extreme Ultraviolet Imager (EUVI) Event Catalog 2006 – 2012

We generated an event catalog with an automated detection algorithm based on the entire EUVI image database observed with the two Solar Terrestrial Relations Observatory (STEREO)-A and -B spacecraft over the first six years of the mission (2006 – 2012). The event catalog includes the heliographic positions of some 20 000 EUV events, transformed from spacecraft coordinates to Earth-based coordinates, and information on associated GOES flare events (down to the level of GOES A5-class flares). The 304 Å wavelength turns out to be the most efficient channel for flare detection (79 % of all EUVI event detections), while the 171 Å (4 %), 195 Å (10 %), and the 284 Å channel (7 %) retrieve substantially fewer flare events, partially due to the suppressing effect of EUV dimming, and partially due to the lower cadence in the later years of the mission. Due to the Sun-circling orbits of STEREO-A and -B, a large number of flares have been detected on the farside of the Sun, invisible from Earth, or seen as partially occulted events. The statistical size distributions of EUV peak fluxes (with a power-law slope of α P =2.5±0.2) and event durations (with a power-law slope of α T =2.4±0.3) are found to be consistent with the fractal-diffusive self-organized criticality model. The EUVI event catalog is available on-line at secchi.lmsal.com/EUVI/euvi_autodetection/euvi_events.txt and may serve as a comprehensive tool to identify stereoscopically observed flare events for 3D reconstruction and to study occulted flare events.

The Critical Group of the Vertex Corona of Cycles and Complete Graphs in Industry

Self-organized criticality is an important theory widely used in various domain such as a variety of industrial accidents, power system, punctuated equilibrium in biology etc.. The Critical Group of the graph is mainly focused on the Abelian sandpile model of self-organized criticality, whose order is the number of spanning trees in the graph, and which is closely connected with the graph Laplacian matrix. In this paper, the main tools will be the computation for Smith normal form of an integer matrix, which can be achieved by the implementation of a series of row and column operations in the ring Ζ of integers. Hence, the structure of the critical group on the vertex corona is determined and it is shown that the Smith normal form is the direct sum of n (m-1)+1cyclic groups. Furthermore, it follows from Kirchooffs Matrix Tree Theorem that the number of spanning trees of the Graph is n (m+1)n (m-1).

Learning more by sampling less: subsampling effects are model specific

When studying real world complex networks, one rarely has full access to all their components. As an example, the central nervous system of the human consists of 1011 neurons which are each connected to thousands of other neurons [1]. Of these 100 billion neurons, at most a few hundred can be recorded in parallel. Thus observations are hampered by immense subsampling. While subsampling does not affect the observables of single neuron activity, it can heavily distort observables which characterize interactions between pairs or groups of neurons [2]. Without a precise understanding how subsampling affects these observables, inference on neural network dynamics from subsampled neural data remains limited. We systematically studied subsampling effects in three self-organized critical (SOC) models, since this class of models can reproduce the spatio-temporal activity of spontaneous activity observed in vivo [2,3]. The models differed in their topology and in their precise interaction rules. The first model consisted of locally connected integrate- and fire units, thereby resembling cortical activity propagation mechanisms [2]. The second model had the same interaction rules but random connectivity [4]. The third model had local connectivity but different activity propagation rules [5]. As a measure of network dynamics, we characterized the spatio-temporal waves of activity, called avalanches. Avalanches are characteristic for SOC models and neural tissue [6]. Avalanche measures A (e.g. size, duration, shape) were calculated for the fully sampled and the subsampled models. To mimic subsampling in the models, we considered the activity of a subset of units only, discarding the activity of all the other units. Under subsampling the avalanche measures A depended on three main factors: First, A depended on the interaction rules of the model and its topology, thus each model showed its own characteristic subsampling effects on A. Second, A depended on the number of sampled sites n. With small and intermediate n, the true A¬ could not be recovered in any of the models. Third, A depended on the distance d between sampled sites. With small d, A was overestimated, while with large d, A was underestimated. Since under subsampling, the observables depended on the model's topology and interaction mechanisms, we propose that systematic subsampling can be exploited to compare models with neural data: When changing the number and the distance between electrodes in neural tissue and sampled units in a model analogously, the observables in a correct model should behave the same as in the neural tissue. Thereby, incorrect models can easily be discarded. Thus, systematic subsampling offers a promising and unique approach to model selection, even if brain activity was far from being fully sampled.

A reaction-diffusion model of cholinergic retinal waves and self-organized criticality

Prior to receiving visual stimuli, spontaneous, correlated activity called retinal waves drives activity-dependent developmental programs. Early-stage waves mediated by acetylcholine (ACh) manifest as slow, spreading bursts of action potentials. They are believed to be initiated by the spontaneous firing of Starburst Amacrine Cells (SACs), whose dense, recurrent connectivity then propagates this activity laterally. Their extended inter-wave intervals and shifting wave boundaries are the result of the slow after-hyperpolarization of the SACs creating an evolving mosaic of recruitable and refractory cells, which can and cannot participate in waves. Recent evidence suggests that cholinergic waves may be modulated by the extracellular concentration of ACh [1].Here, we have constructed a simplified, yet biophysically realistic, reaction-diffusion model of cholinergic retinal waves capable of recapitulating wave dynamics observed in mice retina recordings (Figure ​(Figure1A).1A). The dense, recurrent connectivity of SACs is modeled through local, excitatory coupling occurring via the volume release and diffusion of ACh. This novel approach is used to determine how extracellular ACh may modulate wave activity. In contrast with previous, simulation-based models (e.g. the model of Hennig [2]), we are able to use non-linear wave theory (traveling fronts, pulses, singular perturbation analysis, etc.) to connect wave features to underlying physiological parameters, making our model useful in determining appropriate pharmacological manipulations to experimentally produce waves of a prescribed spatiotemporal character (Figure ​(Figure1B1B). Figure 1 A: Simulated voltage dynamics of model. SACs have Morris-Lecar dynamics, with local coupling occurring via diffusion of extracellular ACh, and a noisy channel to induce spontaneous firing. B: Traveling wave speed C as a function of threshold parameter ... This is the first mathematical analysis of its type on retinal waves. However, a number of theoretical issues remain unresolved. The distribution of wave sizes has been reported to obey a power-law distribution, suggesting the developing retina may exist in a critical state [2]. Are these findings compatible with our theoretical model? We present preliminary results suggesting that our model possesses a configuration in which wave sizes are distributed according to a power-law (Figure ​(Figure1C).1C). We adapt analyses typically used in neural field equations to understand the effects of stochasticity and heterogeneity on wave size statistics [3], and therefore provide theoretical arguments characterizing the potential for criticality in retinal development.

Neuronal avalanches change from wakefulness to deep sleep - a study of intracranial depth recordings in humans

Poster presentation: Twenty Second Annual Computational Neuroscience Meeting: CNS*2013. Paris, France. 13-18 July 2013. Neuronal dynamics differs between wakefulness and sleep stages, so does the cognitive state [1]. In contrast, a single attractor state, called self-organized critical (SOC), has been proposed to govern human brain dynamics for its optimal information coding and processing capabilities [2]. Here we address two open questions: First, does the human brain always operate in this computationally optimal state, even during deep sleep? Second, previous evidence for SOC was based on activity within single brain areas [3-5], however, the interaction between brain areas may be organized differently. Here we asked whether the interaction between brain areas is SOC. We addressed these questions by characterizing neuronal avalanches [3] - spatiotemporal waves of enhanced activity - from up to 61 local field potential (LFP) channels of intracranial depth recordings (5 human patients, two recording nights each, summing up to ~100h of recordings). The recording contacts were distributed inside the entire brain. Note that ~60 contacts are sufficient to avoid major subsampling effects: Subsampling may heavily distort results in SOC systems due to an insufficient number of sampling sites [5]. In addition, we compared the experimental results to results from a subsampled SOC model of integrate- and fire neurons, which can be tuned to the sub- and supercritical regime. We show that avalanche distributions closely follow a power law - the hall mark feature of SOC systems. This result held for each vigilance state, and independent of the threshold and the temporal scale. This indicates first that the interaction between brain areas are close to SOC, and second that the dynamics of all cognitive states, from wakefulness to deep sleep are close to SOC. Minor differences between cognitive states are, however, reflected in the avalanche distributions: Slow wave sleep (s3/s4) showed larger and longer neuronal avalanches than REM sleep, while wakefulness showed intermediate ones (p < 0.05). The SOC neuronal model, together with the data, suggested first that these differences are mediated by global but tiny changes in synaptic strength, and second, that the changes with vigilance states reflect small deviations from criticality to the subcritical regime, implying that the human brain does not operate in the SOC state proper - contrary to previous believes. Independent of criticality, the analysis confirmed that slow wave sleep shows increased correlations between cortical areas, and revealed that REM sleep shows more fragmented cortical dynamics.

DOES A SCALING LAW EXIST BETWEEN SOLAR ENERGETIC PARTICLE EVENTS AND SOLAR FLARES?

Among many other natural processes, the size distributions of solar X-ray flares and solar energetic particle (SEP) events are scale-invariant power laws. The measured distributions of SEP events prove to be distinctly flatter, i.e., have smaller power-law slopes, than those of the flares. This has led to speculation that the two distributions are related through a scaling law, first suggested by Hudson, which implies a direct nonlinear physical connection between the processes producing the flares and those producing the SEP events. We present four arguments against this interpretation. First, a true scaling must relate SEP events to all flare X-ray events, and not to a small subset of the X-ray event population. We also show that the assumed scaling law is not mathematically valid and that although the flare X-ray and SEP event data are correlated, they are highly scattered and not necessarily related through an assumed scaling of the two phenomena. An interpretation of SEP events within the context of a recent model of fractal-diffusive self-organized criticality by Aschwanden provides a physical basis for why the SEP distributions should be flatter than those of solar flares. These arguments provide evidence against a close physical connection of flares with SEP production.

Cycles in mining seismicity

Stochastic models of self-organized criticality and intermittent criticality are used to describe the structure of seismic catalogs. The intermittent models introduce three phases of the seismic cycle: increase in seismic energy, seismic relaxation, and seismic quiescence after the final relaxation. In this paper, seismic mining catalogs from a deep copper mine are searched to find these three phases of the seismic cycle. In spite of the differences between the seismic records from earthquakes and the building of stresses in the mine, the cycles can be estimated in mining seismicity.