Diffusion of information, innovation, and ideas is an important phenomenon in social networks. Information propagates through the network and reaches from one person to the next. In many settings, it is meaningful to restrict diffusion so that each node can spread information to only a limited number of its neighbors rather than to all of them. Such social networks are called closed social networks. In recent years, social media platforms have emerged as an effective medium for commercial entities, where the objective is to maximize profit. In this paper, we study the Profit Maximization in Closed Social Networks (PMCSN) problem in the context of viral marketing. The input to the problem is a closed social network and two positive integers $\ell$ and $B$. The problem asks to select seed nodes within a given budget $B$; during the diffusion process, each node is restricted to choose at most $\ell$ outgoing links for information diffusion; and the objective is to maximize the profit earned by the seed set. The PMCSN problem generalizes the Influence Maximization problem, which is NP-hard. We propose two solution approaches for PMCSN: a sampling-based approximate solution and a marginal-gain-based heuristic solution. We analyze the sample complexity, running time, and space requirements of the proposed approaches. We conduct experiments on real-world, publicly available social network datasets. The results show that the seed sets and diffusion links chosen by our methods yield higher profit than baseline methods. The implementation and data are available at \texttt{https://github.com/PoonamSharma-PY/ClosedNetwork}.

翻译：信息、创新和思想的传播是社交网络中的重要现象。信息通过网络传播，从一个人传递到下一个人。在许多场景中，限制传播是有意义的，即每个节点只能将信息传播给有限数量的邻居，而非全部。此类社交网络被称为封闭社交网络。近年来，社交媒体平台已成为商业实体的有效媒介，其目标是实现利润最大化。本文在病毒式营销的背景下研究封闭社交网络中的利润最大化问题。该问题的输入是一个封闭社交网络以及两个正整数 $\ell$ 和 $B$。问题要求在给定预算 $B$ 内选择种子节点；在传播过程中，每个节点最多只能选择 $\ell$ 条出边进行信息传播；目标是通过种子集获得的利润最大化。PMCSN 问题推广了影响力最大化问题，而后者是 NP 难问题。我们为 PMCSN 提出了两种解决方案：一种基于采样的近似解法和一种基于边际增益的启发式解法。我们分析了所提方法的样本复杂度、运行时间和空间需求。我们在真实世界、公开可用的社交网络数据集上进行了实验。结果表明，我们的方法选择的种子集和传播链路比基线方法产生更高的利润。实现代码和数据可在 \texttt{https://github.com/PoonamSharma-PY/ClosedNetwork} 获取。