Abstract
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 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.
Highlights
In most networks, some vertices are more central than the others
Of a node in a network, where the different motivations lead to different centrality measures that were developed in several areas of science
The most basic centrality measure is the degree of a vertex, which we study in this paper
Summary
Some vertices are more central than the others. To model this intuitive feeling, centrality indices were introduced. Another concept of vertex centrality, the personalization, was introduced in 2003 (see [5]), and is a measure that shows how central an individual is according to a given subset R (group of important people) in a given social network. In [8], the authors introduced group centrality for measures of degree, closeness, and betweenness centrality, which we use in this paper. Freeman [11] realized that despite all of the vertex-centrality indices defined up to that point, there was a need for a normalization which could measure the relative importance of a given vertex in a network and would be based on any chosen centrality index. Following Freeman’s approach, the group centralization notion was introduced in [13], which brings us to the focus of this paper
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