  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
$0\leq\eta<1$, 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 $\tau=(\gamma-2 \eta)/(\gamma-1-\eta)$ and
$\delta=(\gamma-1-\eta)/(\gamma-2)$ for $\gamma<3-\eta$, 3/2 and 2 for
$\gamma>3-\eta$, respectively. The power-law distributions are modified by a
logarithmic correction at $\gamma=3-\eta$.
