  We study the large N expansion of the partition function of the quiver
superconformal Chern-Simons theories deformed by two continuous parameters
which correspond to the general R-charge assignment to the matter fields.
Though the deformation breaks the conformal symmetry, we find that the
partition function shares various structures with the superconformal cases,
such as the Airy function expression of the perturbative expansion in 1/N with
the overall constant A(k) related to the constant map in the ABJM case through
a simple rescaling of k. We also identify five kinds of the non-perturbative
effects in 1/N which correspond to the membrane instantons. The instanton
exponents and the singular structure of the coefficients depend on the
continuous deformation parameters, in contrast to the superconformal case where
all the parameters are integers associated with the orbifold action on the
moduli space. This implies that the singularity of the instanton effects would
be observable also in the gravity side.
