Self-organized criticality and coevolution of network structure and dynamics

We investigate, by numerical simulations, how the avalanche dynamics of the Bak-Tang-Wiesenfeld sandpile model can induce emergence of scale-free networks and how this emerging structure affects dynamics of the system.

Figure 1

Example of rewiring process. A gray colored node started an avalanche of size 2. The number close to each end of the link describes a moment of its last rewiring.

Figure 2

Node degree distributions of rewired network in three time moments: $t_{\mathit{\text{start}}}=0$, $t_{\mathit{\text{mid}}}=2500$, and $t_{\mathit{\text{end}}}=10000$. Inset: the same data binned logarithmically.

Figure 3

Distributions of avalanche area and node degree in time $t_{\mathit{\text{end}}}$. Data are logarithmically binned. Lines are linearly fitted with the values indicated at the figure. A $K\text{−}S$ test gives a $p$ value 0.997 at the level of significance $\alpha =0.05$. In other words, the difference between the samples is not significant.

Figure 4

Process of equilibration of exponents $\tilde{\gamma }$ and $\tau$. Solid line presents theoretical ${\tau }_{\mathit{theor}}$ obtained from $\tilde{\gamma }$ and Eq. (1). Inset: second moment of the degree distribution in time.

Figure 5

Dependence of avalanche area exponent $\tau$ on parameter $\tilde{\gamma }$ of generated scale-free network. The arrow shows direction of a movement in space of parameters during the process of equilibration. Measurements are done in equal time steps $\Delta t=500$ and marked as open circles. The black dot depicts the fixed point of the process. Dashed line presents theoretical ${\tau }_{\mathit{theor}}$ obtained from $\tilde{\gamma }$ and Eq. (1).