Sandpile avalanche dynamics on scale-free networks

Avalanche dynamics is an indispensable feature of complex systems. Here we study the self-organized critical dynamics of avalanches on scale-free networks with degree exponent $\gamma$ through the Bak-Tang-Wiesenfeld (BTW) sandpile model. The threshold height of a node $i$ is set as $k_i^{1-\eta }$ with $0\le \eta <1$, where $k_i$ is the degree of node $i$. Using the branching process approach, we obtain the avalanche size and the duration distribution of sand toppling, which follow power-laws with exponents $\tau$ and $\delta$, respectively. They are given as $\tau =(\gamma -2\eta )\mathrm{/}(\gamma -1-\eta )$ and $\delta =(\gamma -1-\eta )\mathrm{/}(\gamma -2)$ for $\gamma <3-\eta$, 3/2 and 2 for $\gamma >3-\eta$, respectively. The power-law distributions are modified by a logarithmic correction at $\gamma =3-\eta$.