  We study how the Roberge-Weiss (RW) transition depends on the pattern of
center symmetry breaking using a $\mathbb{Z}_{3}$-QCD model. We adopt
flavor-dependent quark imaginary chemical potentials, namely
$(\mu_{u},\mu_{d},\mu_{s})/iT=(\theta-2\pi{C}/3,\,\theta,\,\theta+2\pi{C}/3)$
with $C\in[0,1]$. The RW periodicity is guaranteed and the center symmetry of
$\mathbb{Z}_{3}$-QCD is explicitly broken when $C\neq{1}$ or/and quark masses
are non-degenerate. For $N_{f}=3$ and $C\neq{1}$, the RW transition occurs at
$\theta=\theta_{RW}=(2k+1)\pi/3\,(k\in\mathbb{Z})$, which becomes stronger with
decrease of $C$. When $C={1}$, the $\theta_{RW}$ turns into $2k\pi/3$ for
$N_{f}=2+1$, but keeps $(2k+1)\pi/3$ for $N_{f}=1+2$; in both cases, the RW
transitions get stronger with the mass mismatch. For other $C\neq{0}$ cases,
the $\theta_{RW}$'s are not integral multiples of $\pi/3$. We find that the RW
transition is more sensitive to the deviation of $C$ from one compared to the
mass non-degeneracy and thus the strength of the traditional RW transition with
$C=0$ is the strongest. The nature of RW endpoints and its implications to
deconfinement transition are investigated.
