Sandpile on Scale-Free Networks

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 τ. Applying the theory of the 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 a uniform threshold, finding that the two exponents reduce to the mean-field ones.