Bak-Tang-Wiesenfeld Model Research Articles

Self-organized Speculation Game for the spontaneous emergence of financial stylized facts

We sophisticate the Speculation Game to relax the need for rigorous tuning of a part of parameters for the emergence of characteristic statistics of price returns. Named as Self-organized Speculation Game (SOSG), thanks to the concept of the Bak–Tang–Wiesenfeld (BTW) sandpile model, the model can spontaneously adjust the number of market participants during the process of reaching the quasi-critical state. While the original Speculation Game’s high reproducibility of financial stylized facts is maintained, the behavioral characteristics of this modified model, such as the dynamics of system size and the observations of power-law phenomena, strongly resemble those of the BTW model. The market size is evolutionally settled with fluctuations within a certain range, similar to the total sand grains behave in the sandpile model. SOSG infers the possibility that, in the real financial markets, it could be the self-organized quasi-criticality which works behind the spontaneous emergence of universal financial stylized facts.

Critical properties of deterministic and stochastic sandpile models on two-dimensional percolation backbone

Deterministic Bak Tang Wiesenfeld (BTW) model and stochastic sandpile model (SSM), a modified Manna model, are studied on the percolation backbone, a random fractal with well known fractal features, generated on a square lattice in 2-dimensions. In spite of the underlying random structure of the backbone, the BTW model preserves its positive time auto-correlation and multifractal behaviour due to its complete toppling balance. In contrast, the critical properties of the SSM still exhibits finite size scaling (FSS) as it manifests on the regular lattices. Various scaling relations are developed analysing the topography of the avalanches. The extended set of critical exponents obtained for the SSM is found to obey various scaling relations in terms of the fractal dimension dfB of the backbone, on the other hand, the BTW model does not exhibit any such scaling relations. As the critical exponents of the SSM defined on the backbone depend on dfB, they are found to be entirely different from those known for the SSM defined on the regular lattice as well as on deterministic fractals. The SSM on the percolation backbone is found to obey FSS and belongs to a new universality class.

Impact of network topology on self-organized criticality.

The general mechanisms behind self-organized criticality (SOC) are still unknown. Several microscopic and mean-field theory approaches have been suggested, but they do not explain the dependence of the exponents on the underlying network topology of the SOC system. Here, we first report the phenomena that in the Bak-Tang-Wiesenfeld (BTW) model, sites inside an avalanche area largely return to their original state after the passing of an avalanche, forming, effectively, critically arranged clusters of sites. Then, we hypothesize that SOC relies on the formation process of these clusters, and present a model of such formation. For low-dimensional networks, we show theoretically and in simulation that the exponent of the cluster-size distribution is proportional to the ratio of the fractal dimension of the cluster boundary and the dimensionality of the network. For the BTW model, in our simulations, the exponent of the avalanche-area distribution matched approximately our prediction based on this ratio for two-dimensional networks, but deviated for higher dimensions. We hypothesize a transition from cluster formation to the mean-field theory process with increasing dimensionality. This work sheds light onto the mechanisms behind SOC, particularly, the impact of the network topology.

Bak-Tang-Wiesenfeld model in the upper critical dimension: Induced criticality in lower-dimensional subsystems.

We present extensive numerical simulations of Bak-Tang-Wiesenfeld (BTW) sandpile model on the hypercubic lattice in the upper critical dimension D_{u}=4. After re-extracting the critical exponents of avalanches, we concentrate on the three- and two-dimensional (2D) cross sections seeking for the induced criticality which are reflected in the geometrical and local exponents. Various features of finite-size scaling (FSS) theory have been tested and confirmed for all dimensions. The hyperscaling relations between the exponents of the distribution functions and the fractal dimensions are shown to be valid for all dimensions. We found that the exponent of the distribution function of avalanche mass is the same for the d-dimensional cross sections and the d-dimensional BTW model for d=2 and 3. The geometrical quantities, however, have completely different behaviors with respect to the same-dimensional BTW model. By analyzing the FSS theory for the geometrical exponents of the two-dimensional cross sections, we propose that the 2D induced models have degrees of similarity with the Gaussian free field (GFF). Although some local exponents are slightly different, this similarity is excellent for the fractal dimensions. The most important one showing this feature is the fractal dimension of loops d_{f}, which is found to be 1.50±0.02≈3/2=d_{f}^{GFF}.

Bak–Tang–Wiesenfeld model on the square site-percolation lattice

The Bak–Tang–Wiesenfeld (BTW) model is considered on the site-diluted square lattice, tuned by the occupancy probability p. Various statistical observables of the avalanches are analyzed in terms of p, e.g. the fractal dimension of their exterior frontiers, gyration radius, loop lengths and Green’s function. The model exhibits critical behavior for all amounts of p, and the exponents of the statistical observables are analyzed. We find a distinct universality class at , which is unstable towards a p = 1 (BTW) fixed point. This universality class displays some common features such as a two-dimensional (2D) Ising universality class, e.g. the fractal dimension of loops in the thermodynamic limit is which is compatible with the fractal dimension of geometrical spin clusters of the 2D critical Ising model (with ).

Bak–Tang–Wiesenfeld model in the finite range random link lattice

We consider the BTW model in random link lattices with finite range interaction (RLFRI). The degree distribution for nodes is considered to be uniform in the interval $(0,n_0)$. We numerically calculate the exponents of the distribution functions in terms of $(n_0,R)$ in which $R$ is the range of interactions. Dijkstra radius is utilized to calculate the fractal dimension of the avalanches. Our analysis shows that there is, at least one length scale ($r_0(n_0,R)$) in which the fractal dimension is changed. We find that for the scales smaller than $r_0(n_0,R)$, which is typically one decade, the fractal dimension is nearly independent of $n_0$ and $R$ and is equal to $1.4$, i.e. close to that of the BTW in the regular lattice ($1.25$). Using this fact and other analysis, we conclude that the BTW-type behaviors are dominant for small values of $n_0$ and $R$, whereas for large values of these parameters a new regime is seen in which the exponent of distribution function of avalanche masses is nearly $1.4$. We also numerically calculate the explicit form of the \textit{number of unstable nodes} (NUN) as a time dependent process and show that for regular lattice it is (up to a normalization) proportional to a one dimensional Weiner process and for RLFRI it acquires a drift term. Using this dynamical variable it is numerically shown that we can not continuously approach the regular lattice limit by decreasing $R$.

Activity-dependent branching ratios in stocks, solar x-ray flux, and the Bak-Tang-Wiesenfeld sandpile model

We define an activity-dependent branching ratio that allows comparison of different time series X(t). The branching ratio b(x) is defined as b(x)=E[xi(x)/x]. The random variable xi(x) is the value of the next signal given that the previous one is equal to x, so xi(x)=[X(t+1) | X(t)=x]. If b(x)>1, the process is on average supercritical when the signal is equal to x, while if b(x)<1, it is subcritical. For stock prices we find b(x)=1 within statistical uncertainty, for all x, consistent with an "efficient market hypothesis." For stock volumes, solar x-ray flux intensities, and the Bak-Tang-Wiesenfeld (BTW) sandpile model, b(x) is supercritical for small values of activity and subcritical for the largest ones, indicating a tendency to return to a typical value. For stock volumes this tendency has an approximate power-law behavior. For solar x-ray flux and the BTW model, there is a broad regime of activity where b(x) approximately equal 1, which we interpret as an indicator of critical behavior. This is true despite different underlying probability distributions for X(t) and for xi(x). For the BTW model the distribution of xi(x) is Gaussian, for x sufficiently larger than 1, and its variance grows linearly with x. Hence, the activity in the BTW model obeys a central limit theorem when sampling over past histories. The broad region of activity where b(x) is close to one disappears once bulk dissipation is introduced in the BTW model-supporting our hypothesis that it is an indicator of criticality.

Scale-free vortex cascade emerging from random forcing in a strongly coupled system

We elucidate the unifying aspects of self-organized criticality (SOC) and turbulence through analysis of data from a laboratory dusty plasma monolayer. We compare analysis of experimental data with simulations of a two-dimensional (2D) many-body system, of 2D chaotic fluid flow, and two different SOC-models, the Zhang and the Bak–Tang–Wiesenfeld (BTW) models, all subject to steady random forcing at small scales. The scale-free vortex cascade is apparent from structure functions as well as spatio-temporal avalanche analysis. We find similar scaling exponents for the experiment, the many-body simulation, and the fluid simulation, indicating some common dynamical features. However, the exponents of the Zhang model are different from those of the BTW model, and they are all different from those of the dust and fluid systems. Thus, we conclude that the dust monolayer dynamics can be viewed as turbulent as well as avalanching, but a fluid model is a better representation of the dust dynamics for this particular experiment than the sandpile models considered. The experiment exhibits global fluctuation statistics consistent with a recent hypothesis predicting universal non-Gaussian probability density functions, but the model systems yield this result only in a restricted range of forcing conditions.

Avalanche exponents and corrections to scaling for a stochastic sandpile

We study distributions of dissipative and nondissipative avalanches in Manna's stochastic sandpile, in one and two dimensions. Our results lead to the following conclusions: (1) avalanche distributions, in general, do not follow simple power laws, but rather have the form P(s) approximately s(-tau(s))(ln s)(gamma)f(s/s(c)), with f a cutoff function; (2) the exponents for sizes of dissipative avalanches in two dimensions differ markedly from the corresponding values for the Bak-Tang-Wiesenfeld (BTW) model, implying that the BTW and Manna models belong to distinct universality classes; (3) dissipative avalanche distributions obey finite-size scaling, unlike in the BTW model.

Proof of breaking of self-organized criticality in a nonconservative abelian sandpile model

By computer simulations, it was reported that the Bak-Tang-Wiesenfeld (BTW) model loses self-organized criticality (SOC) when some particles are annihilated in a toppling process in the bulk of system. We give a rigorous proof that the BTW model loses SOC as soon as the annihilation rate becomes positive. To prove this, a nonconservative Abelian sandpile model is defined on a square lattice, which has a parameter alpha (>/=1) representing the degree of breaking of the conservation law. This model is reduced to be the BTW model when alpha=1. By calculating the average number of topplings in an avalanche <T> exactly, it is shown that for any alpha>1, <T><infinity even in the infinite-volume limit. The power-law divergence of <T> with an exponent 1 as alpha-->1 gives a scaling relation 2nu(2-a)=1 for the critical exponents nu and a of the distribution function of T. The 1-1 height correlation C11(r) is also calculated analytically and we show that C11(r) is bounded by an exponential function when alpha>1, although C11(r) approximately r(-2d) was proved by Majumdar and Dhar for the d-dimensional BTW model. A critical exponent nu(11) characterizing the divergence of the correlation length xi as alpha-->1 is defined as xi approximately |alpha-1|(-nu(11)) and our result gives an upper bound nu(11)</=1/2.

A deterministic sandpile automaton revisited

The Bak-Tang-Wiesenfeld (BTW) sandpile model is a cellular automaton which has been intensively studied during the last years as a paradigm for self-organized criticality. In this paper, we reconsider a deterministic version of the BTW model introduced by Wiesenfeld, Theiler and McNamara, where sand grains are added always to one fixed site on the square lattice. Using the Abelian sandpile formalism we discuss the static properties of the system. We present numerical evidence that the deterministic model is only in the BTW universality class if the initial conditions and the geometric form of the boundaries do not respect the full symmetry of the square lattice.