Statistical investigation of avalanches of three-dimensional small-world networks and their boundary and bulk cross-sections

In many situations we are interested in the propagation of energy in some portions of a three-dimensional system with dilute long-range links. In this paper, a sandpile model is defined on the three-dimensional small-world network with real dissipative boundaries and the energy propagation is studied in three dimensions as well as the two-dimensional cross-sections. Two types of cross-sections are defined in the system, one in the bulk and another in the system boundary. The motivation of this is to make clear how the statistics of the avalanches in the bulk cross-section tend to the statistics of the dissipative avalanches, defined in the boundaries as the concentration of long-range links ($\alpha$) increases. This trend is numerically shown to be a power law in a manner described in the paper. Two regimes of $\alpha$ are considered in this work. For sufficiently small $\alpha \mathrm{s}$ the dominant behavior of the system is just like that of the regular BTW, whereas for the intermediate values the behavior is nontrivial with some exponents that are reported in the paper. It is shown that the spatial extent up to which the statistics is similar to the regular BTW model scales with $\alpha$ just like the dissipative BTW model with the dissipation factor (mass in the corresponding ghost model) $m^2\sim \alpha$ for the three-dimensional system as well as its two-dimensional cross-sections.

Figure 1

(a) Schematic set up of a small-world network with some random long-range links. Note that the regular bonds are not altered. (b) Two schematic microavalanches that are connected by a single link between two points, $i$ and $j$. The gray sites are the toppled sites, and the red lines are the exterior frontiers of these microavalanches. Note that the point, e.g., $i$ for the right-hand microavalanche, plays the role of a sink point. This is not, however, a perfect picture, since this point can play the role of the grain source as well. (c) The three-dimensional system with an avalanche and its projection in its cross-section in $z_{\text{cross-section}}=L/2$.

Figure 2

(a) The plot of $M_3$ in terms of $R_3$ and the corresponding exponents ${\gamma }_{M_3{R}_3}$. Upper inset: ${\gamma }_{M_3{R}_3}^{\text{UV}}$ and ${\gamma }_{M_3{R}_3}^{\text{IR}}$. Lower inset: the crossover radius $R_3^{*}$ in terms of $\alpha$ with the exponent $0.5\pm 0.03$. (b) The same as (a) for various lattice sizes $L$ (These curves have been shifted by a constant for more clearness). The finite-size-dependent (UV and IR) slopes ${\gamma }_{M_3{R}_3}$ have been shown in the inset. (c) The log-log plot of the distribution function $P\left(R_3\right)$ in terms of $R_3$). Inset: the crossover point $R_3^{*}$. (d) Schematic representation of the confined area between the distribution functions of $M_3$ for bulk and boundary injections. (e) The IR exponent of the distribution function of $R_3$ for bulk (${\tau }_2$) and boundary $\left[{\tau }_2(z=0)\right]$ injections in terms of $\alpha$. The corresponding confined area has been shown in the inset.

Figure 3

The log-log plot of the distribution function of (a) $M_2$ and (b) $r$ for various rates of $\alpha$. Insets: the power-law behavior of the confined area of (a) $C_{M_2}$ and $M_2^{*}$ and (b) $C_r$ and $r^{*}$ in terms of $\alpha$.

Figure 4

(a) The fractal dimension ${\gamma }_{M_2{R}_2}$. Upper inset: the exponents for $z=L/2$ and $z=0$ cross-sections. Lower inset: spatial extents of regular BTW behavior for $z=L/2$ and $z=0$ cross-sections. (b) The fractal dimension ${\gamma }_{lr}$ (The curves have been shifted by a constant for more clearness). Inset: The exponents for $z=L/2$ and $z=0$ cross-sections in terms of $\alpha$. (c) The finite size dependence of the exponents. (d) The hyperscaling exponents ${\gamma }_{x,y}\equiv \frac{{\tau }_y-1}{{\tau }_x-1}$ and the calculated exponents in terms of $\alpha$.