  Inhomogeneous random matrices with a non-trivial variance profile determined
by a symmetric stochastic matrix and with independent symmetrically
sub-Gaussian entries up to Hermitian symmetry, include many prominent examples,
such as remarkable sparse Wigner matrices and random band matrices in dimension
d, and have been of great interest recently. The maximum of entry variances as
a natural proxy of sparsity is both a key feature and a significant obstacle.
In this paper, we study low-rank additive perturbations of such random
matrices, and establish a sharp BBP phase transition for extreme eigenvalues at
the level of law of large numbers. Under suitable conditions on the variance
profile and the finite-rank perturbation, we also establish the fluctuations of
spectral outliers that may be characterized by the general inhomogeneous random
matrices. This reveals the strong non-universality phenomena, which may depend
on eigenvectors, sparsity or geometric structure. Our proofs rely on ribbon
graph expansions, upper bounds for diagram functions, large moment estimates
and counting of typical diagrams.
