Bak Tang Wiesenfeld Research Articles

Load cascades in spatial networks: A sandpile model approach

In the real world, the construction of infrastructure networks is influenced by spatial factors. For such complex networks, We refer to them as spatial networks. This study investigates, based on the Bak–Tang–Wiesenfeld sandpile model, the impact of spatial factors on sandpile cascading avalanches. Through simulations, we discover that spatial factors not only facilitate the occurrence of moderate-scale avalanches in the system but also have a suppressing effect on the occurrence of the largest cascading avalanches. Using a multiplicative branching process, we analyze the influence of triangular structures composed of nodes on sandpile avalanches and find that nodes within triangular structures are relatively resistant to topples. Because of the relatively stable nature within triangular structures, more load tends to stay within this kind of structures. Once a cascading avalanche occurs in the system, the avalanche scale is often larger than that in the non-spatial counterpart. This increase in avalanche scale results in more load being released by the system, making the spatial networks more relaxed. And their probability of very-large-scale cascading avalanches is lower than that of non-spatial networks.

Near universal values of social inequality indices in self-organized critical models

We have studied few social inequality measures associated with the sub-critical dynamical features (measured in terms of the avalanche size distributions) of four self-organized critical models while the corresponding systems approach their respective stationary critical states. It has been observed that these inequality measures (specifically the Gini and Kolkata indices) exhibit nearly universal values though the models studied here are widely different, namely the Bak–Tang–Wiesenfeld sandpile, the Manna sandpile and the quenched Edwards–Wilkinson interface, and the fiber bundle interface. These observations suggest that the self-organized critical systems have broad similarity in terms of these inequality measures. A comparison with similar earlier observations in the data of socio-economic systems with unrestricted competitions suggest the emergent inequality as a result of the possible proximity to the self-organized critical states.

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.

Predictability and Scaling in a BTW Sandpile on a Self-similar Lattice

This paper explores the predictability of a Bak–Tang–Wiesenfeld isotropic sandpile on a self-similar lattice, introducing an algorithm which predicts the occurrence of target events when the stress in the system crosses a critical level. The model exhibits the self-organized critical dynamics characterized by the power-law segment of the size-frequency event distribution extended up to the sizes $$\sim L^{\beta }$$ , $$\beta = \log _3 5$$ , where L is the lattice length. We establish numerically that there are large events which are observed only in a super-critical state and, therefore, predicted efficiently. Their sizes fill in the interval with the left endpoint scaled as $$L^{\alpha }$$ and located to the right from the power-law segment: $$\alpha \approx 2.24 > \beta $$ . The right endpoint scaled as $$L^3$$ represents the largest event in the model. The mechanism of predictability observed with isotropic sandpiles is shown here for the first time.

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.

Power-law correlations and coupling of active and quiet states underlie a class of complex systems with self-organization at criticality.

Physical and biological systems often exhibit intermittent dynamics with bursts or avalanches (active states) characterized by power-law size and duration distributions. These emergent features are typical of systems at the critical point of continuous phase transitions, and have led to the hypothesis that such systems may self-organize at criticality, i.e. without any fine tuning of parameters. Since the introduction of the Bak-Tang-Wiesenfeld (BTW) model, the paradigm of self-organized criticality (SOC) has been very fruitful for the analysis of emergent collective behaviors in a number of systems, including the brain. Although considerable effort has been devoted in identifying and modeling scaling features of burst and avalanche statistics, dynamical aspects related to the temporal organization of bursts remain often poorly understood or controversial. Of crucial importance to understand the mechanisms responsible for emergent behaviors is the relationship between active and quiet periods, and the nature of the correlations. Here we investigate the dynamics of active (θ-bursts) and quiet states (δ-bursts) in brain activity during the sleep-wake cycle. We show the duality of power-law (θ, active phase) and exponential-like (δ, quiescent phase) duration distributions, typical of SOC, jointly emerge with power-law temporal correlations and anti-correlated coupling between active and quiet states. Importantly, we demonstrate that such temporal organization shares important similarities with earthquake dynamics, and propose that specific power-law correlations and coupling between active and quiet states are distinctive characteristics of a class of systems with self-organization at criticality.

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 ).

Controlling the self-organizing dynamics in a sandpile model on complex networks by failure tolerance

In this paper, we propose a strategy to control the self-organizing dynamics of the Bak-Tang-Wiesenfeld (BTW) sandpile model on complex networks by allowing some degree of failure tolerance for the nodes and introducing additional active dissipation while taking the risk of possible node damage. We show that the probability for large cascades significantly increases or decreases respectively when the risk for node damage outweighs the active dissipation and when the active dissipation outweighs the risk for node damage. By considering the potential additional risk from node damage, a non-trivial optimal active dissipation control strategy which minimizes the total cost in the system can be obtained. Under some conditions the introduced control strategy can decrease the total cost in the system compared to the uncontrolled model. Moreover, when the probability of damaging a node experiencing failure tolerance is greater than the critical value, then no matter how successful the active dissipation control is, the total cost of the system will have to increase. This critical damage probability can be used as an indicator of the robustness of a network or system.

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 $R$.

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.

Controlling Self-Organizing Dynamics on Networks Using Models that Self-Organize

Controlling self-organizing systems is challenging because the system responds to the controller. Here, we develop a model that captures the essential self-organizing mechanisms of Bak-Tang-Wiesenfeld (BTW) sandpiles on networks, a self-organized critical (SOC) system. This model enables studying a simple control scheme that determines the frequency of cascades and that shapes systemic risk. We show that optimal strategies exist for generic cost functions and that controlling a subcritical system may drive it to criticality. This approach could enable controlling other self-organizing systems.

Suppressing cascades of load in interdependent networks

Understanding how interdependence among systems affects cascading behaviors is increasingly important across many fields of science and engineering. Inspired by cascades of load shedding in coupled electric grids and other infrastructure, we study the Bak-Tang-Wiesenfeld sandpile model on modular random graphs and on graphs based on actual, interdependent power grids. Starting from two isolated networks, adding some connectivity between them is beneficial, for it suppresses the largest cascades in each system. Too much interconnectivity, however, becomes detrimental for two reasons. First, interconnections open pathways for neighboring networks to inflict large cascades. Second, as in real infrastructure, new interconnections increase capacity and total possible load, which fuels even larger cascades. Using a multitype branching process and simulations we show these effects and estimate the optimal level of interconnectivity that balances their trade-offs. Such equilibria could allow, for example, power grid owners to minimize the largest cascades in their grid. We also show that asymmetric capacity among interdependent networks affects the optimal connectivity that each prefers and may lead to an arms race for greater capacity. Our multitype branching process framework provides building blocks for better prediction of cascading processes on modular random graphs and on multitype networks in general.

Natural time analysis of critical phenomena

A quantity exists by which one can identify the approach of a dynamical system to the state of criticality, which is hard to identify otherwise. This quantity is the variance κ(1)(≡<χ(2)> - <χ>(2)) of natural time χ, where <f(χ)> = Σp(k)f(χ(k)) and p(k) is the normalized energy released during the kth event of which the natural time is defined as χ(k) = k/N and N stands for the total number of events. Then we show that κ(1) becomes equal to 0.070 at the critical state for a variety of dynamical systems. This holds for criticality models such as 2D Ising and the Bak-Tang-Wiesenfeld sandpile, which is the standard example of self-organized criticality. This condition of κ(1) = 0.070 holds for experimental results of critical phenomena such as growth of rice piles, seismic electric signals, and the subsequent seismicity before the associated main shock.

Variable predictability in deterministic dissipative sandpile

Abstract. It is known that some quiescence precedes the strong events in the Bak–Tang–Wiesenfeld sand-pile (Pepke and Carlson, 1994) We introduce dissipation depending on the propagation of the events into this model such that in the constructed model the growth of activity occurs before the strong events. This fact allows the prediction of them in advance with a certain efficiency. This efficiency is variable in time. The best predictability is observed during subcritical time ranges, while the efficiency is definitely worse in the supercritical state.

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.

A stochastic theory for temporal fluctuations in self-organized critical systems

A stochastic theory for the toppling activity in sandpile models is developed, based on a simple mean-field assumption about the toppling process. The theory describes the process as an anti-persistent Gaussian walk, where the diffusion coefficient is proportional to the activity. It is formulated as a generalization of the Itô stochastic differential equation with an anti-persistent fractional Gaussian noise source and a deterministic drift term. An essential element of the theory is rescaling to obtain a proper thermodynamic limit. When subjected to the most relevant statistical tests, the signal generated by the stochastic equation is indistinguishable from the temporal features of the toppling process obtained by numerical simulation of the Bak–Tang–Wiesenfeld sandpile.

CELL-TO-SITE RENORMALIZATION GROUP STUDY OF A SANDPILE

This study investigates the Bak–Tang–Wiesenfeld sandpile using a renormalization group (RG) approach based on the similarity of L × L RG cells of different scales. The fundamental difficulty of this approach arises from the lack of full enumeration of all relaxations inside a RG cell as L ≥ 3. This study develops a simple sampling algorithm to sample the relaxations and applies this algorithm to the RG calculations for L = 3, 4, and 8. As the fixed point analysis shows, increasing L does not lead the resultant height probabilities toward the exact solution.

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.