Coevolutionary extremal dynamics on gasket fractal

Abstract

We considered a Bak–Sneppen model on a Sierpinski gasket fractal. We calculated the avalanche size distribution and the distribution of distances between subsequent minimal sites. To observe the temporal correlations of the avalanche, we estimated the return time distribution, the first-return time, and the all-return time distribution. The avalanche size distribution follows the power law, P ( s ) ∼ s − τ , with the exponent τ = 1.004 ( 7 ) . The distribution of jumping sites also follows the power law, P ( r ) ∼ r − π , with the critical exponent π = 4.12 ( 4 ) . We observe the periodic oscillation of the distribution of the jumping distances which originated from the jumps of the level when the minimal site crosses the stage of the fractal. The first-return time distribution shows the power law, P f ( t ) ∼ t − τ f , with the critical exponent τ f = 1.418 ( 7 ) . The all-return time distribution is also characterized by the power law, P a ( t ) ∼ t − τ a , with the exponent τ a = 0.522 ( 4 ) . The exponents of the return time satisfy the scaling relation τ f + τ a = 2 for τ f ⩽ 2 .