Sandpile on Scale-Free Networks

Abstract

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