Branching Process Approach to Avalanche Dynamics on Complex Networks

We investigate the avalanche dynamics of the Bak-Tang-Wiesenfeld (BTW) sandpile model on complex networks with general degree distributions. With the threshold height of each node given as its degree in the model, self-organized criticality emerges such that the avalanche size and the duration distribution follow power laws with exponents τ and δ, respectively. Applying the theory of the multiplicative branching process, we find that the exponents τ and δ are given as τ = γ/(γ− 1) and δ = (γ − 1)/(γ − 2) for the degree distribution pd(k) ∼ k−γ with 2 < γ < 3, with a logarithmic correction at γ = 3, while they are 3/2 and 2, respectively, for γ > 3 and when pd(k) follows an exponential-type distribution. The analytic solutions are supported by our numerical simulation results.