  We study the task of selecting $k$ nodes, in a social network of size $n$, to
seed a diffusion with maximum expected spread size, under the independent
cascade model with cascade probability $p$. Most of the previous work on this
problem (known as influence maximization) focuses on efficient algorithms to
approximate the optimal seed set with provable guarantees given knowledge of
the entire network; however, obtaining full knowledge of the network is often
very costly in practice. Here we develop algorithms and guarantees for
approximating the optimal seed set while bounding how much network information
is collected. First, we study the achievable guarantees using a sublinear
influence sample size. We provide an almost tight approximation algorithm with
an additive $\epsilon n$ loss and show that the squared dependence of sample
size on $k$ is asymptotically optimal when $\epsilon$ is small. We then propose
a probing algorithm that queries edges from the graph and use them to find a
seed set with the same almost tight approximation guarantee. We also provide a
matching (up to logarithmic factors) lower-bound on the required number of
edges. This algorithm is implementable in field surveys or in crawling online
networks. Our probing takes $p$ as an input which may not be known in advance,
and we show how to down-sample the probed edges to match the best estimate of
$p$ if they are collected with a higher probability. Finally, we test our
algorithms on an empirical network to quantify the tradeoff between the cost of
obtaining more refined network information and the benefit of the added
information for guiding improved seeding strategies.
