Average avalanche size <i>ES</i> as a function of the system loading level <i>L</i>.

<p>Results for five different restoration strategies: (1) <i>p_r</i> = 0 (black diamonds); (2) <i>t_r</i> = 1, <i>p_r</i> = 1, = 6×10⁻⁶, = 8×10⁻⁶ (green squares); (3) <i>t_r</i> = 4, <i>p_r</i> = 1, = 6×10⁻⁶, = 8×10⁻⁶ (purple triangles); (4) <i>t_r</i> = 1, <i>p_r</i> = 1, = −8×10⁻⁶, = −6×10⁻⁶ (red stars); (5) <i>t_r</i> = 4, <i>p_r</i> = 1, = −8×10⁻⁶, = −6×10⁻⁶ (blue circles). Each curve corresponds to the average over twenty thousand realizations of networks with 10⁵ components. The example network system has no specific topology, on which the results do not depend. The initial component loading can vary from <i>L</i>_min to <i>L</i>_max = <i>L</i>_fail = 1. Then, <i>L</i> = (<i>L</i>_min+1)/2 may be increased by increasing <i>L</i>_min. The initial disturbance <i>D = </i> 4×10⁻⁶ is assumed to be the same as the load transfer amount <i>P = </i>4×10⁻⁶. All the investigated network systems without restoration satisfy the cascading condition that the cascade step is no less than 5.