Sandpile on Scale-Free Networks

Abstract

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 $τ$. Applying the theory of multiplicative branching process, we obtain the exponent $τ$ and the dynamic exponent $z$ as a function of the degree exponent $γ$ of SF networks as $τ=γ/(γ-1)$ and $z=(γ-1)/(γ-2)$ in the range $2 < γ< 3$ and the mean field values $τ=1.5$ and $z=2.0$ for $γ>3$, with a logarithmic correction at $γ=3$. 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.