  Influence maximization (IM) aims at maximizing the spread of influence by
offering discounts to influential users (called seeding). In many applications,
due to user's privacy concern, overwhelming network scale etc., it is hard to
target any user in the network as one wishes. Instead, only a small subset of
users is initially accessible. Such access limitation would significantly
impair the influence spread, since IM often relies on seeding high degree
users, which are particularly rare in such a small subset due to the power-law
structure of social networks. In this paper, we attempt to solve the limited IM
in real-world scenarios by the adaptive approach with seeding and diffusion
uncertainty considered. Specifically, we consider fine-grained discounts and
assume users accept the discount probabilistically. The diffusion process is
depicted by the independent cascade model. To overcome the access limitation,
we prove the set-wise friendship paradox (FP) phenomenon that neighbors have
higher degree in expectation, and propose a two-stage seeding model with the FP
embedded, where neighbors are seeded. On this basis, for comparison we
formulate the non-adaptive case and adaptive case, both proven to be NP-hard.
In the non-adaptive case, discounts are allocated to users all at once. We show
the monotonicity of influence spread w.r.t. discount allocation and design a
two-stage coordinate descent framework to decide the discount allocation. In
the adaptive case, users are sequentially seeded based on observations of
existing seeding and diffusion results. We prove the adaptive submodularity and
submodularity of the influence spread function in two stages. Then, a series of
adaptive greedy algorithms are proposed with constant approximation ratio.
