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$k$，公平影响力最大化问题的目标是选择一个种子集合$S$（$|S|\leq k$），在最大化影响力传播的同时缩小不同社区之间的影响力差距。尽管存在多种公平概念，但福利公平概念（平衡了公平性与影响力传播）展现出显著的有效性。然而，缺乏用于优化福利公平目标函数的高效算法，限制了其仅能应用于仅有数百节点的小规模网络。本文采用福利公平的目标函数，最大化所有社区受影响比例的指数加权求和。我们首先引入算术平均分数幂的无偏估计量。然后，通过适配反向影响力采样（RIS）方法，将优化问题转化为加权最大覆盖问题。我们还分析了高概率逼近公平影响力所需的反向可达集合数量。此外，我们提出了一种高效的算法，保证$1-1/e-\varepsilon$的近似比。