  We study the quenched invariance principle for random conductance models with
long range jumps on $\Z^d$, where the transition probability from $x$ to $y$
is, on average, comparable to $|x-y|^{-(d+\alpha)}$ with $\alpha\in (0,2)$ but
is allowed to be degenerate. Under some moment conditions on the conductance,
we prove that the scaling limit of the Markov process is a symmetric
$\alpha$-stable L\'evy process on $\R^d$.The well-known corrector method in
homogenization theory does not seem to work in this setting. Instead, we
utilize probabilistic potential theory for the corresponding jump processes.
Two essential ingredients of our proof are the tightness estimate and the
H\"{o}lder regularity of caloric functions for non-elliptic
$\alpha$-stable-like processes on graphs. Our method is robust enough to apply
not only for $\Z^d$ but also for more general graphs whose scaling limits are
nice metric measure spaces.
