  For real world systems, nonuniform medium is ubiquitous. Therefore, we
investigate the diffusion-limited-aggregation process on a two dimensional
directed small-world network instead of regular lattice. The network structure
is established by rewiring connections on the two dimensional directed lattice.
Those rewired edges are controlled by two parameters $\theta$ and $m$, which
characterize the spatial length and the density of the long-range connections,
respectively. Simulations show that there exists a maximum value of the fractal
dimension when $\theta$ equals zero. Interestingly, we find that the symmetry
of the aggregation pattern is broken when rewired connections are long enough,
which may be an explanation for the formation of asymmetrical fractal in
nature. Then, we perform multifractal analysis on the patterns further.
