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Self-Organized Cooperative Criticality in Coupled Complex Systems

We show that the coupled complex systems can evolve into a new kind of self-organized critical state where each subsystem is not critical, however, they cooperate to be critical. This criticality is different from the classical BTW criticality where the single system itself evolves into a critical state. We also find that the outflows can be accumulated in the coupled systems. This will lead to the emergency of spatiotemporal intermittency in the critical state.

Since proposed in 1980s, the self-organized criticality (SOC) [1][2][3] has been one of the most popular theory in complex science and is widely used as a way of understanding emergent complex behavior in physical [4][5][6][7][8], biological [9][10][11] and even social systems [12][13][14][15]. According to this theory, the ubiquitous power law in nature can be interpreted as the hallmark of critical state, a stable state that systems evolve into by themselves. The SOC is usually illustrated by a simple cellular automaton, the so-called BTW sandpile model [1][2][3]. The dynamics of this model can be imaged as a transport phenomenon, where sand grains in a very high heap (with a local height gradient exceeding the threshold) is unstable and will tumble down the slope of this heap (against the local height gradient). We note that this dynamics is not enough to describe the real complex systems where a movement along the local gradient is also possible. For example, in our daily life you may have considered to migrate to a bigger city for a better job. Migration to a bigger city is a movement along the population gradient. Here, someone's final choice is determined by the equilibrium between the population gradient and other gradients, such as the job gradients, the education gradients and even shopping gradients. In other words, the dynamics of migration is coupled with other dynamics. In fact, similar coupling also exists in the natural systems. A well-known example is the cross effect in the transport arising in a mixture if both the concentrations and the temperature are non-uniform over the system. The cross effect can be simply illustrated in a isotropic binary mixture without viscosity when no external forces are supposed to be present and the pressure is uniform over the system [16]: Symbols J q , J d , T , C and ρ are the heat flux, the diffusion flow, the temperature, the concentration of one of the components in the mixture and the density of mixture respectively. The diffusion coefficient D and the heat conductivity λ are positive and are related to the normal * liulei@mail.iap.ac.cn heat conduction and diffusion respectively. The symbol s T is called Soret coefficient which is related to the cross effect in the transport and can be negative or positive. The coefficients α ≥ 0 and β ≥ 0. From Eqs. (1) and (2), one can see that the direction of heat or diffusion flow is determined by the equilibrium between the local temperature and the local concentration gradient. Two transport processes are coupled together. From above discussions, a question arises as to what will happen if the dynamics of two SOC systems are coupled together?
In this paper, we try to answer this question by using a simple model. The model we proposed here is composed of two BTW sandpiles. A BTW sandpile is represented by a twodimensional grid with a length of N . At each square of the grid with coordinates (x, y), we assign a number z(x, y), which represents the negative value of local height gradient at that square. The coordinates 1 ≤ x ≤ N and 1 ≤ y ≤ N . Note that what z represents in the sandpile model is not important. The more important thing is its dynamics. In real systems, z would represent any interesting dynamical variable (see examples in Refs. [4][5][6][7][8][9][10][11][12][13][14][15]). Different from the dynamics of BTW sandpile, our model couple two sandpiles in the way that if and only if both the negative value of local height gradients of z1 and z2 of the two sandpiles at the same positions (x,y) are greater than the threshold z c (equaling 3 in the following simulations), the sand grains then tumble down, and the tumbledown grains are transported to the nearest-neighbor grids, z1, z2(x ± 1, y) = z1, z2(x ± 1, y) + 1 (4) z1, z2(x, y ± 1) = z1, z2(x, y ± 1) + 1 (5) This model is conservative except in the boundary, for example, Two sandpiles are statistically symmetric and we just show the statistics of one of them in the following analysis. Self-Organized Cooperative Criticality (SOCC). This is the most important feature we found in the coupled sandpiles. Starting from two sandpiles with the same length of N (equaling 50 in the following simulations) and the zero local height gradients in the whole sandpiles, the critical states are built by adding sand grains randomly to each sandpile simultaneously according to the rule: which is called the nonconservative perturbation in the BTW sandpile model [17]. After each perturbation, the two coupled sandpiles are allowed to tumble down (if needed) according to Eqs. (3), (4) and (5). A local perturbation would spread to the nearest-neighbor grids. These grids would tumble down again and the perturbation is transported to the next-nearest-neighbor grids and so on until the unstable grids (i.e. z1, z2(x, y) > z c ) are not found in the sandpiles. Then another perturbations are randomly added into each sandpile to trigger a new possible avalanche. As time elapses, the coupled sandpiles will evolve into a self-organized critical state. A signal for the critical state is the outflow which is defined as the total number of sand grains flowing out of one sandpile after each perturbation. Both for the BTW sandpile (blue broken line in Fig. 1a) and for the coupled sandpiles (red broken line in Fig. 1a), the outflow will fluctuate around a constant once the critical states are built. There are other signals for the critical state, such as the distributions of the avalanche size (also called the total energy release) and the lifetime of avalanche [1][2][3]17]. The avalanche size s is defined as the total number of tumbles in one sandpile induced by each perturbation and the lifetime of avalanche t is defined as the total number of simultaneous tumbles in one sandpile induced by each perturbation. In the simulation, perturbations are not simultaneously added into the two sandpiles and we will randomly choose a sandpile for each perturbation. We count the avalanche size s and the lifetime t induced by each perturbation and find that in the critical states the distribution of avalanche sizes D(s) and the distribution of lifetimes D(t) follow the same power laws both in the coupled sandpiles and the BTW sandpile ( Fig. 1b and 1c).
Although the coupled sandpiles and the BTW sandpile have the same distributions of avalanche sizes and lifetimes, their self-organized critical states are very different in nature. For the BTW sandpile, all z(x, y) ≤ z c in the critical state. For the coupled sandpiles, there are many "supcritical" grids with z1 > z c or z2 > z c which mean that each separate sandpile in the coupled model is not critical in view of the BTW sandpile. However, they are indeed in the self-organized critical state not separately but cooperatively. Cooperative phenomena are common in the complex systems. Many species, such as bees and ants, can behave cooperatively in a selforganized way (see Ref. [18] and references therein). Our model join the two important notions of the cooperation and the SOC together in a simple manner. Thus, we call the new critical mode the self-organized cooperative criticality (SOCC).
The left plot of Fig. 2 shows a snapshot of the structure of minimally stable state with "z1 = z c and z2 > z c " or "z1 > z c and z2 = z c " in one sandpile. It likes a fractal structure and its dimension D can be measured as follows [19]. We lay a square in the center of whole sandpile and count the number of minimally stable states n in this square. Then, a power-law relation between the length of the square L and n can be set up. That is the length-number relation: n ∼ L D , where D is called the  cluster fractal dimension. Measured result shows that D = log n/ log L ≈ 2 (the right plot of Fig. 2), which means that the structure of minimally stable state in the coupled sandpiles is a fat fractal [20]. More interesting, the structure of minimally stable state (with z = z c ) in the BTW sandpile is also a fat fractal with the same cluster fractal dimension D ≈ 2. Thus, the geometry structures of minimally stable state are statistically similar both in the SOC and SOCC sandpiles. That is possibly why we obtain the similar distributions of avalanche size and lifetimes in the two models. Accumulative Effect and Spatiotemporal Intermittency. We note that the outflow in the SOC state fluctuates around a value of 1 (the blue broken line in Fig. 1a). This means that the inflow and the outflow in the SOC state is in statistical equilibrium and the average of z(x, y) will also fluctuate around a constant (the blue lines in Fig. 1a). However, in the SOCC state the outflow fluctuates around a constant smaller than 1 (the red broken line in Fig. 1a). This means that the inflow is greater than the outflow and the adding grains will be accumulated in the sandpile in a constant rate (the red line in Fig. 1a).
In Fig. 3a, we analyze the averages of z with z > z c and z ≤ z c respectively and find that the adding grains are mainly accumulated to the "supcritical" grids with z < z c . To see the accumulative process intuitively, we observe the evolution of the distribution of z in the diagonal direction in one sandpile (Fig. 3b). One can see that as time elapses the grids with z > z c accumulate more and more adding grains. At the same time, a more and more intermittent distribution of z appears in space (Fig. 3b). The intermittency will also appear in the time series of z at a fixed grid (Fig. 3c). In all, we observe the spatiotemporal intermittency of local height gradients in the coupled sandpiles. Many complex systems has the spatiotemporal intermittency, such as the fluid turbulence [21][22][23], the earthquakes [3,24] and the traffic jam [14]. The accumulative effect of the SOCC sandpiles provides another possible mechanism for the emergency of spatiotemporal intermittency.
Robustness. To test the robustness of the criticality in the coupled model, we randomly remove the connections between nearest-neighbor grids in each sandplie. When the connection between nearest-neighbor grids, say z1(x, y) and z1(x+1, y), is removed, the rules of Eqs. (3), (4) and (5) are modified by In the simulation, the numbers of removed connections for the two sandpiles are the same. Figures 4a and 4b show the distributions of D(s) and D(t) with different  numbers of removed connections. We find that these distributions also have the power laws and no changes have been found in the exponents. However, the statistics on the very large avalanches deviate from the power laws.
With the increase of removal, the deviation becomes large. The connectivity of the sandpile is decreased by the removal. Thus, the large avalanches occur with a less frequency than in the unspoiled sandpile. We also remove the correlations between the two sandpiles at the same coordinations. At these grids without correlations, if z1 > z c or z2 > z c the sand grains will tumble down. Simulations show that almost nothing is changed in the distributions of D(s) and D(t) even the proportion of the removed correlations is increased to 20% (Figs. 4c and 4d ).
Non-conservative Coupled Sandpiles. The average of z in the conservative coupled sandpiles will increase infinitely if we continually add sand grains into the sandpiles. It seems unphysical. In real systems, the dissipation always exists. In Ref. [25], authors proposed a nonconservative BTW sandpile to simulate the effect of dissipation. In this model, when tumbles happen at (x, y), no matter how large z(x, y) is, it will become zero. Following this sprit, we modify the rules of (3) by z1, z2(x, y) = 0. Other rules like Eqs. (4) and (5) are not changed.
The non-conservative coupled sandpiles will also evolve into the SOCC states. Different from the conservative sandpiles, the inflow and outflow in this SOCC state is in statistical equilibrium. It means that the accumulation effect is statistically offset by the dissipation. We stress "statistically" here because the local grids with z > z c will also accumulate sand grains if tumbles have not happened in these grids. We measure the distributions of D(s) and D(t) in the critical states and find that they also have the power laws with the same exponents as in the conservative BTW sandpile (Figs. 5b and 5c).
In conclusion, we propose a model by coupling two BTW sandpiles in a simple way. This model can conjoin many notions popular in the complex science, including the SOC, the cooperation, and the spatiotemporal intermittency. We believe that our model could be further extended to understand the real systems which might be composed of many coupled SOC systems.