Figure 4. Transition from a spanning tree to a square lattice by adding bonds.

On the left a spanning tree is shown (blue arrows). The dark blue nodes indicate the external boundary and the red nodes the internal boundary (IB). Initially bonds are added (red-dashed arrows) at the IB reducing the size of the IB and introducing loops into the network. In this configuration 50% of the IB is removed by adding edges. On the right the avalanche size distributions obtained during this process are shown. On the spanning tree the distribution scales with an exponent of 1.5. Completely removing the internal boundary is already sufficient to obtain an exponent close to 1.28 and adding more bonds until the complete square lattice (SQL) is obtained does not change the exponent significantly. The inset shows that the distributions, obtained on a network where 50% of the IB is removed, do not scale with the system size.