  We study a family online influence maximization problems where in a sequence
of rounds $t=1,\ldots,T$, a decision maker selects one from a large number of
agents with the goal of maximizing influence. Upon choosing an agent, the
decision maker shares a piece of information with the agent, which information
then spreads in an unobserved network over which the agents communicate. The
goal of the decision maker is to select the sequence of agents in a way that
the total number of influenced nodes in the network. In this work, we consider
a scenario where the networks are generated independently for each $t$
according to some fixed but unknown distribution, so that the set of influenced
nodes corresponds to the connected component of the random graph containing the
vertex corresponding to the selected agent. Furthermore, we assume that the
decision maker only has access to limited feedback: instead of making the
unrealistic assumption that the entire network is observable, we suppose that
the available feedback is generated based on a small neighborhood of the
selected vertex. Our results show that such partial local observations can be
sufficient for maximizing global influence. We model the underlying random
graph as a sparse inhomogeneous Erd\H{o}s--R\'enyi graph, and study three
specific families of random graph models in detail: stochastic block models,
Chung--Lu models and Kronecker random graphs. We show that in these cases one
may learn to maximize influence by merely observing the degree of the selected
vertex in the generated random graph. We propose sequential learning algorithms
that aim at maximizing influence, and provide their theoretical analysis in
both the subcritical and supercritical regimes of all considered models.
