# Early Warning Signals for Critical Transitions in Sandpile Cellular Automata

^{1}Laboratory of Complex Systems Modeling and Control, National Research University Higher School of Economics, Moscow, Russia^{2}Cybersecurity Research Center, University of Bernardo O’Higgins, Santiago, Chile^{3}Department of Business Informatics, National Research University Higher School of Economics, Moscow, Russia

The sandpile cellular automata, despite the simplicity of their basic rules, are adequate mathematical models of real-world systems, primarily open nonlinear systems capable to self-organize into the critical state. Such systems surround us everywhere. Starting from processes at microscopic distances in the human brain and ending with large-scale water flows in the oceans. The detection of critical transitions precursors in sandpile cellular automata will allow progress significantly in the search for effective early warning signals for critical transitions in complex real systems. The presented paper is devoted to the detection and investigation of such signals based on multifractal analysis of the time series of falls of the cellular automaton cells. We examined cellular automata in square lattice and random graphs using standard and facilitated rules. It has been established that log wavelet leaders cumulant are effective early warning measures of the critical transitions. Common features and differences in the behavior of the log cumulants when cellular automata transit into the self-organized critical state and the self-organized bistability state are also established.

## Introduction

Open complex systems usually operate in a nonequilibrium state, which can lead to the appearance of fluctuations in them, induced by external influence. When the initial structureless state is lost, which is an extrapolation of the equilibrium state to nonequilibrium conditions, a critical transition occurs in the system, leading to the emergence of new stationary states. In addition to the specified critical transition that occurs as a result of bifurcations (the so-called bifurcation-induced critical transition), noise-induced critical transition and rate-induced critical transition can occur in systems. An important feature of such critical transitions is the fact that such transitions have common features, despite the differences in the details of the elements interactions of each system. Due to this reason, many common (unifying) quantitative and qualitative precursors of critical transitions or early warning signals (EWS) in the critical transitions have been proposed (see the papers [1–5]). Despite this, we assume that there should be differences in the EWS for different types of critical transitions, at least in the neighborhood of the critical transition point. Finding such differences is one of the objectives of our research.

The justification for the use of most early warning measures is associated with an increase in the time that needed to return to a stable state with small disturbances in the neighborhood of the critical point. These EWS include autocorrelation, variance, skewness and kurtosis, power spectral density, and Hurst exponents. These measures are estimated for the time series characterizing changes in some parameters of the systems. For example, the order parameter can be used as such parameter, if the critical transition is the first or the second order phase transition. Other EWS are recurrence measures such as determinism, laminarity and entropy (see the paper [6]).

Complex systems surround us everywhere. Starting from processes at microscopic distances in the human brain and ending with large-scale water flows in the oceans. In the complex systems, the interaction of individual elements with each other is so complex that the entire system acquires completely new and unexpected properties that cannot be reduced to the properties of individual parts. Controlling such parameters as temperature or magnetization, it is possible to provide a phase transition—a transition through a critical point, which is characterized by power laws. However, there are various examples of processes and systems (see the papers [7–9]), which are characterized by power laws that have arisen without any parameters’ tunning: seismic activity with destructive earthquakes, neural and social networks, financial markets, forest fires, etc. P. Bak, C. Tang, and K. Wiesenfeld [8] discovered self-organized criticality (SOC) phenomena in 1987. They built a mechanism that explains how a system reaches the critical state without tunning of any parameters. Their model, called the sandpile or BTW model, is implemented on a square lattice on which grains of sand fall. Sandpile cellular automata have simple rules that lead to complex critical behavior. A detailed description of sandpile models is provided in *Time Series Data Generation using Sandpile Cellular Automata*. The self-organized critical transition corresponds to the second order phase transitions. It was recently found (see the papers [10–13]) that in real complex systems the self-organized bistable (SOB) transition is possible, which corresponds to the first order phase transition. A sandpile cellular automata with facilitated rules has also been proposed (see the paper [13]), which is capable to demonstrate the SOB transition. At this moment, we are not aware of papers that present the results of the study of time series features generated by systems when they approach the SOC state and the SOB state. To close this gap, we conducted a study on discovering the EWS of the critical transitions and the features of the critical transitions for sandpile cellular automata. Our study is based on the results of multifractal time series analysis generated by the automata. Research results are presented in this paper.

The paper is structured as follows. *Methods* provides descriptions of local sandpile cellular automata rules in square lattice and random graphs—time series generators for the number of collapsed cells, and the wavelet leader method for time series analysis. In the *Result and Their Discussion*, the results of EWS detection for the critical transitions and multifractal features of the automata being in the subcritical phase and the critical state are presented and discussed. The *Result and Their Discussion* is devoted to the discussion of obtained results, as well as the discussion of possible practical applications obtained by EWS for detecting critical transitions.

## Methods

This Section describes the rules for the operation of sandpile cellular automata - time series generators for number of the falls (

### Time Series Data Generation Using Sandpile Cellular Automata

To date, isotropic sandpile cellular automata (SCA) with a variety of local rules have been developed (see the reviews [14, 15]). To generate the time series data, we used the standard rules of the Bak—Tang—Wiesenfeld (BTW) [8], Feder—Feder (FF) [16] and Manna (M) [17] models. SCA with standard rules (SR) are capable to self-organize into a critical state. We also looked at facilitated sandpile cellular automata or facilitated rules (FR) automata. Such automata are capable to self-organize into a bistable state. A modification of the Manna model as a facilitated SCA model is presented in the paper [13]. Finally, we investigated the dynamics of sand grains not only on square lattice (SL), but also on random networks grown using the Erdos—Renyi (ER) model and the Barabasi—Albert (BA) model (see the papers [18, 19]). The introduced abbreviations will further be used to denote a cellular automaton. For example, FF-ST-BA matches sandpile cellular automata with standard Feder—Feder rules on Barabasi—Albert (BA) network.

The basic operating principle of any SCA is quite simple. Let us describe it in the form of an algorithm, at each step of which the similarities and differences of each of the automata are indicated. First of all, if cellular automata on random graphs are considered, then it is necessary to grow these graphs. A description of cultivation is provided at the end of this Subsection.

Step 1. Randomly selected cells (x, y) of a square lattice or a grown random graph are filled randomly, one particle at a time. As a result, the number of particles in these cells is *z _{i}* (

*x,y*)→

*z*(

_{i}*x,y*)+1.

Step 2. The critical value of particles (*z _{c}*) is determined for each cell. For square lattices

*z*= 4, for random graphs

_{c}*z*is equal to the number of connections of the vertex (

_{c}*x, y*).

Step 3. Collapse of cells and redistribution of particles between cells.

The stability condition for each of the cells of the automaton for a model with standard rules is checked. If *z _{i}* (

*x, y*) ≥

*z*, then the given cell (

_{c}*x, y*) crumbles with the distribution of particles into neighboring cells. After the cell is overturned, grains of sand are distributed equally to each neighboring cell in deterministic models (FF- and BTW-model); a grain of sand falls into a randomly selected neighboring cell in the stochastic M-model. On nodes with degree 1 of random graphs, the collapse of a cell (node) can only lead to the escape of particles from these nodes.

For the model with facilitated rules, the stability condition for each of the cells of the automaton is also checked. If *z _{i}* (

*x, y*) ≥

*z*or

_{c}*f*(

_{i-1}*x, y*) ≥2 (f is the number of falls into a cell on the previous move), then cell (

*x, y*) collapses into neighboring cells in accordance with rules of the model. Also, shedding can be deterministic and random.

Step 4. The number of collapsed cells is calculated, which corresponds to the value of the time series at a certain step.

In conservative models (M- and BTW-model) on the SL, when the unstable cell is overturned, the value in it decreases by the value

The sandpile cellular automata on the RSL are very approximate models of real systems, which are characterized by self-organization into a critical and bistable states. First of all, the approximation of the models is associated with a fixed and limited number of nearest neighbors of each node of the automaton. Therefore, the study of critical transitions in SCA on ER- and BA-networks is under particular interest. For example, although the ER model does not reproduce some of the typical properties of real networks, on average, the model is a good model for transportation networks, contagion and diffusion (see the paper [19]). The BA model is a good model for complex networks, and therefore has a much wider application area (see the papers [20, 21]). Random graph

Below, we consider the formal rules of all studied sandpile cellular automata.

##### Sandpile Cellular Automata on the Square Lattice

Lets

Then the formal rules for BTW-ST-SL automata will take the following form.

BTW-FA-SL automata.

FF-ST-SL automata.

FF-FA-SL automata.

M-ST-SL automata.

M-FA-SL automata.

##### Sandpile Cellular Automata on the Random Graphs

Let

Then the formal rules for BTW-ST automata will take the following form.

BTW-FA automata.

FF-ST automata.

FF-FA automata.

M-ST automata.

M-FA automata.

### Wavelet Leaders Multifractal Analysis of Time Series Generated by Self-Organizing Cellular Automata

Multifractal analysis, being a method of local investigation of the temporal structure of a signal, allows to evaluate its correlation properties even with a relatively short signal registration. This is an undoubted advantage of this method (see the papers [22–24]). There are several methods of multifractal time series analysis, which have their own capabilities and limitations. The most common are multifractal detrended fluctuation analysis (MFDFA) [25, 26], wavelet transform maxima modules (WTMM) [27], and wavelet leader method (WLM) [28, 29], which is the development of the WTMM method. We used the WLM to estimate the multifractal singularity spectrum. One of the obvious advantages of WLM in relation to the MFDF method is the absence of the need to detrend the initial time series data, because the wavelets are not sensitive to the trend. In addition, the MFDFA method gives good estimates only for positive values of Holder exponents; at the same time, the accuracy of determining the values of

Without consideration technical details of WLM, let us consider the main features of the method in the context of the multifractal formalism. A detailed description of the method is presented in the papers [30–32]. After the discrete wavelet transform, the time series is decomposed into discrete wavelet coefficients of different levels, which are presented in the form of a matrix. After that, this matrix is analyzed: the coefficient and its neighbors (right and left) are analyzed at each level. The largest of them is selected. Thus, a set of the largest coefficients is obtained. These are the wavelet leaders defined for each wavelet expansion level.

Next, the standard procedure for multifractal analysis will be considered. Structural functions are found that have the following form:

where

The scaling exponent

where *i*th log-cumulant.

The singularity spectrum

where

Expressions (14) and (15) allow to represent

If

Thus, the doublet

## Results and Their Discussion

In this Section, we consider the features of the time series for number of falls (

### Long-Range Dependence in the Time Series for the Sandpile Cellular Automata

Figure 1 shows the time series *x _{t}*,

*t*∈ [0,10000], for the BTW automata on the square lattice. These figures also show the autocorrelation functions and Holder exponents (

**FIGURE 1**. Time series of falls for a standard **(left)** and facilitated **(right)** BTW automaton, and the corresponding autocorrelation functions (log-log plot) and Holder exponents.

It has been established (Figure 1) that for all cellular automata the rate of decrease of the autocorrelation function in the SubC phase

The value

Thus, the first two log cumulants can be used as precursors of critical transitions in the sandpile cellular automata. The next Subsection is devoted to a discussion of this problem.

### Early Detection of Critical Transitions Based on Time Series for Log Wavelet Leaders Cumulant

In a previous Subsection, we showed that

Figure 2 shows the time series

**FIGURE 2**. Time series of the log cumulants for the standard **(left)** and facilitated **(right)** BTW automaton.

All cellular automata show the same behavior

## Discussion

This Section presents a discussion of linking of our research results to some recent results from the theory of early warning indicators for critical transitions. Also, a discussion of possible practical applications of proposed early warning measures for detection of critical transitions is presented.

We will start by considering the main similarities and differences in the stochastic dynamics of the number of unstable cells *Result and Their Discussion*. Common to automata with standard and facilitated rules is the multifractal structure

As it is shown in the paper [35], the average magnetization time series in the Ising model are multifractal in the SubC phase and in the critical state. Moreover, the structure of the time series is more heterogeneous in the critical state, than in the SubC phase. The increase of the memory in time series as the system approaches the critical point is also shown. In our opinion, a significant difference between the indicated results and those obtained by us lies in the fundamental difference between the rules for the Ising model and the rules for the sandpile cellular automata. In the paper [36], the results of the study of changes in the temporal autocorrelation at lag 1 and the power spectral density (PSD) as the system approaches to the critical point are presented. There was a significant increase in the autocorrelation in the neighborhood of the critical point, which is associated with the critical slowing down, as well as an increase in the parameter

The time series generated by real systems have a more complex structure than the time series generated by the sandpile cellular automata. However, we believe that the main features of

Finally, we look at the limitations and possible further research in the analysis of critical transition precursors in sandy cellular automata.

We have established only one limitation of the proposed approach associated with the length of the analyzed time series and the large number of zero values in it. To obtain reliable results, the length of the time series must be at least 2000 steps. Otherwise, significant fluctuations in the value of the log cumulants are observed, in which it is impossible to determine their smoothed behavior.

From the point of view of the prospects for further research, in our opinion, one should focus on the interpretation of jumps in the values of log-cumulants, which are characteristic of critical states of all cellular automata. An explanation of this empirical phenomenon is possible by analyzing the time series obtained under various initial conditions and sizes of all investigated cellular automata.

## Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

## Author Contributions

AD, VK, and VD designed the research. VK, VD and NA performed the simulations. NA prepared the figures. All authors contributed to writing and reviewing the manuscript.

## Funding

The work is an output of a research project implemented as part of the Basic Research Program at the National Research University Higher School of Economics (HSE University).

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

## Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2022.839383/full#supplementary-material

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Keywords: early warning signals, sandpile cellular automata, self-organized criticality, selforganized bistability, wavelet leaders method, log-cumulants, multifractal formalism

Citation: Dmitriev A, Kornilov V, Dmitriev V and Abbas N (2022) Early Warning Signals for Critical Transitions in Sandpile Cellular Automata. *Front. Phys.* 10:839383. doi: 10.3389/fphy.2022.839383

Received: 19 December 2021; Accepted: 10 January 2022;

Published: 31 January 2022.

Edited by:

Ming Li, Zhejiang University, ChinaReviewed by:

Alexander Shapoval, University of Łódź, PolandVictor Popov, Lomonosov Moscow State University, Russia

Copyright © 2022 Dmitriev, Kornilov, Dmitriev and Abbas. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Andrey Dmitriev, a.dmitriev@hse.ru