Dissipative stochastic sandpile model on small world network : properties of non-dissipative and dissipative avalanches

A dissipative stochastic sandpile model is constructed and studied on small world networks in one and two dimensions with different shortcut densities $\phi$, where $\phi=0$ represents regular lattice and $\phi=1$ represents random network. The effect of dimension, network topology and specific dissipation mode (bulk or boundary) on the the steady state critical properties of non-dissipative and dissipative avalanches along with all avalanches are analyzed. Though the distributions of all avalanches and non-dissipative avalanches display stochastic scaling at $\phi=0$ and mean-field scaling at $\phi=1$, the dissipative avalanches display non trivial critical properties at $\phi=0$ and $1$ in both one and two dimensions. In the small world regime ($2^{-12} \le \phi \le 0.1$), the size distributions of different types of avalanches are found to exhibit more than one power law scaling with different scaling exponents around a crossover toppling size $s_c$. Stochastic scaling is found to occur for $s<s_c$ and the mean-field scaling is found to occur for $s>s_c$. As different scaling forms are found to coexist in a single probability distribution, a coexistence scaling theory on small world network is developed and numerically verified.