Given a graph $G$, a community structure $\mathcal{C}$, and a budget $k$, the fair influence maximization problem aims to select a seed set $S$ ($|S|\leq k$) that maximizes the influence spread while narrowing the influence gap between different communities. While various fairness notions exist, the welfare fairness notion, which balances fairness level and influence spread, has shown promising effectiveness. However, the lack of efficient algorithms for optimizing the welfare fairness objective function restricts its application to small-scale networks with only a few hundred nodes. In this paper, we adopt the objective function of welfare fairness to maximize the exponentially weighted summation over the influenced fraction of all communities. We first introduce an unbiased estimator for the fractional power of the arithmetic mean. Then, by adapting the reverse influence sampling (RIS) approach, we convert the optimization problem to a weighted maximum coverage problem. We also analyze the number of reverse reachable sets needed to approximate the fair influence at a high probability. Further, we present an efficient algorithm that guarantees $1-1/e - \varepsilon$ approximation.

翻译：给定图$G$、社区结构$\mathcal{C}$和预算$k$，公平影响最大化问题旨在选择种子集$S$（$|S|\leq k$），在最大化影响传播的同时缩小不同社区之间的影响差距。尽管存在多种公平性概念，但福利公平性概念（平衡公平程度与影响传播）已展现出显著的有效性。然而，由于缺乏优化福利公平性目标函数的高效算法，该方法仅能应用于节点数仅数百的小规模网络。本文采用福利公平性目标函数，最大化所有社区受影响比例的指数加权求和。我们首先提出算术平均值的分数次幂的无偏估计量，随后通过适配反向影响采样（RIS）方法，将优化问题转化为加权最大覆盖问题。我们还分析了以高概率近似公平影响所需的反向可达集数量，并进一步提出一种保证$1-1/e-\varepsilon$近似比的高效算法。