Sandpile on Scale-Free Networks

We investigate the avalanche dynamics of the Bak-Tang-Wiesenfeld (BTW) sandpile model on scale-free (SF) networks, where threshold height of each node is distributed heterogeneously, given as its own degree. We find that the avalanche size distribution follows a power law with an exponent $\tau$. Applying the theory of multiplicative branching process, we obtain the exponent $\tau$ and the dynamic exponent $z$ as a function of the degree exponent $\gamma$ of SF networks as $\tau =\gamma \mathrm{/}(\gamma -1)$ and $z=(\gamma -1)\mathrm{/}(\gamma -2)$ in the range $2<\gamma <3$ and the mean field values $\tau =1.5$ and $z=2.0$ for $\gamma >3$, with a logarithmic correction at $\gamma =3$. The analytic solution supports our numerical simulation results. We also consider the case of uniform threshold, finding that the two exponents reduce to the mean field ones.