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A dissipative stochastic sandpile model is constructed and studied on smallworld networks in one and two dimensions with different shortcut densities$\phi$, where $\phi=0$ represents regular lattice and $\phi=1$ representsrandom network. The effect of dimension, network topology and specificdissipation mode (bulk or boundary) on the the steady state critical propertiesof non-dissipative and dissipative avalanches along with all avalanches areanalyzed. Though the distributions of all avalanches and non-dissipativeavalanches 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 ofavalanches are found to exhibit more than one power law scaling with differentscaling exponents around a crossover toppling size $s_c$. Stochastic scaling isfound 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 singleprobability distribution, a coexistence scaling theory on small world networkis developed and numerically verified.

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