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.

翻译：我们研究的是在一个规模为$n的社交网络中选择美元节点的任务,以在独立级联模型下以级联概率为美元进行最大预期扩展规模的传播。以前关于该问题的大部分工作(称为影响力最大化)侧重于高效的算法,以最优种子集为准值,以整个网络的知识为背景。然而,在实践中,获得对网络的充分知识是非常昂贵的。为了解决这一差距,我们首先用美元(n)的影响力样本来研究可实现的保证。我们提供一种接近的算法,以接近的美元(1美元/e)的预期扩展规模,使用级联的概率为美元。我们提供一个紧凑的 $1美元(OPT_\\\\\\\\\\ eepsilon n$)的保证,使用 $(k) eqour kreality rqolorizal- silview liverational as the serview liverations the weroughtal rough the weral ral ral-ral-ral ligal) subs the the liver liver liver subs to the we folverd thes the werviews subs the slations the slations lievations liverdaltiquest ds to the slupluptalds to thes to the sluptals to thes) liversaldaltialtials to the slations)