Seeding with Costly Network Information

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 the knowledge of the entire network. However, in practice, obtaining full knowledge of the network is very costly. To address this gap, we first study the achievable guarantees using $o(n)$ influence samples. We provide an approximation algorithm with a tight $(1-1/e){\mbox{OPT}}-\epsilon n$ guarantee, using $O_{\epsilon}(k^2\log n)$ influence samples and show that this dependence on $k$ is asymptotically optimal. We then propose a probing algorithm that queries ${O}_{\epsilon}(p n^2\log^4 n + \sqrt{k p} n^{1.5}\log^{5.5} n + k n\log^{3.5}{n})$ 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. To address the dependence of our probing algorithm on the independent cascade probability $p$, we show that it is impossible to maintain the same approximation guarantees by controlling the discrepancy between the probing and seeding cascade probabilities. Instead, we propose to down-sample the probed edges to match the seeding cascade probability, provided that it does not exceed that of probing. Finally, we test our algorithms on real world data to quantify the trade-off between the cost of obtaining more refined network information and the benefit of the added information for guiding improved seeding strategies.