The importance of individuals and groups in networks is modeled by various centrality measures. Additionally, Freeman’s centralization is a way to normalize any given centrality or group centrality measure, which enables us to compare individuals or groups from different networks. In this paper, we focus on degree-based measures of group centrality and centralization. We address the following related questions: For a fixed k, which k-subset S of members of G represents the most central group? Among all possible values of k, which is the one for which the corresponding set S is most central? How can we efficiently compute both k and S? To answer these questions, we relate with the well-studied areas of domination and set covers. Using this, we first observe that determining S from the first question is $\mathcal{NP}$-hard. Then, we describe a greedy approximation algorithm which computes centrality values over all group sizes k from 1 to n in linear time, and achieve a group degree centrality value of at least $(1-1/e)(w^{*}-k)$, compared to the optimal value of $w^{*}$. To achieve fast running time, we design a special data structure based on the related directed graph, which we believe is of independent interest.

中文翻译：

---

网络中的群组度中心化和集中化

个人和群体在网络中的重要性通过各种集中度度量来建模。另外，Freeman的集中化是一种规范任何给定集中度或组集中度度量的方法，它使我们能够比较来自不同网络的个人或组。在本文中，我们集中于基于程度的小组集中度和集中度度量。我们解决以下相关问题：对于固定k，G成员的k个子集S代表最中心的组？在k的所有可能值中，哪个是对应集合S最中心的那个？我们如何有效地计算k和S？为了回答这些问题，我们与研究充分的统治领域和固定范围相关。使用这个，我们首先观察到从第一个问题确定S是$\mathcal{NP}$-硬。然后，我们描述一种贪心近似算法，该算法在线性时间内计算从1到n的所有组大小k的中心度值，并实现至少一个组度中心值$（\mathrm{1个}-\mathrm{1个}/Ë）（w^{*}-ķ）$，与的最佳值相比 $w^{*}$。为了实现快速运行，我们基于相关的有向图设计了特殊的数据结构，我们认为这是独立的利益。