To promote viral marketing, major social platforms (e.g., Facebook Marketplace and Pinduoduo) repeatedly select and invite different users (as seeds) in online social networks to share fresh information about a product or service with their friends. Thereby, we are motivated to optimize a multi-stage seeding process of viral marketing in social networks and adopt the recent notions of the peak and the average age of information (AoI) to measure the timeliness of promotion information received by network users. Our problem is different from the literature on information diffusion in social networks, which limits to one-time seeding and overlooks AoI dynamics or information replacement over time. As a critical step, we manage to develop closed-form expressions that characterize and trace AoI dynamics over any social network. For the peak AoI problem, we first prove the NP-hardness of our multi-stage seeding problem by a highly non-straightforward reduction from the dominating set problem, and then present a new polynomial-time algorithm that achieves good approximation guarantees (e.g., less than 2 for linear network topology). To minimize the average AoI, we also prove that our problem is NP-hard by properly reducing it from the set cover problem. Benefiting from our two-side bound analysis on the average AoI objective, we build up a new framework for approximation analysis and link our problem to a much simplified sum-distance minimization problem. This intriguing connection inspires us to develop another polynomial-time algorithm that achieves a good approximation guarantee. Additionally, our theoretical results are well corroborated by experiments on a real social network.

翻译：为促进病毒式营销，主流社交平台（如Facebook Marketplace和拼多多）会反复选择并邀请不同用户（作为种子节点）在在线社交网络中向其好友分享关于某产品或服务的新鲜信息。这促使我们优化社交网络中病毒式营销的多阶段种子选取过程，并采用最新提出的信息峰值年龄与平均年龄（AoI）概念来衡量网络用户接收推广信息的时效性。我们的问题不同于现有社交网络信息传播文献——这些研究局限于一次性种子选取，忽视了AoI动态变化或信息随时间更替的现象。作为关键步骤，我们成功推导出能够表征并追踪任意社交网络中AoI动态变化的闭式表达式。针对峰值AoI问题，我们首先通过从支配集问题出发的高度非直接规约，证明了多阶段种子选取问题的NP-hard性；随后提出一种能在线性网络拓扑等场景下实现良好近似比（例如小于2）的新型多项式时间算法。为最小化平均AoI，我们通过将问题适当规约为集合覆盖问题，同样证明了其NP-hard性。得益于对平均AoI目标函数的双侧边界分析，我们构建了新的近似分析框架，并将原问题简化为更易处理的距离和最小化问题。这一有趣关联启发我们开发了另一种具有良好近似保证的多项式时间算法。此外，我们在真实社交网络上的实验充分验证了理论结果的有效性。