Next Article in Journal
An Optimal Derivative-Free Ostrowski’s Scheme for Multiple Roots of Nonlinear Equations
Previous Article in Journal
A New Continuous-Discrete Fuzzy Model and Its Application in Finance

Open AccessArticle

# Group Degree Centrality and Centralization in Networks

by Matjaž Krnc 1,† and Riste Škrekovski 1,2,3,*,†
1
FAMNIT, University of Primorska, 6000 Koper, Slovenia
2
Faculty of Information Studies, 8000 Novo Mesto, Slovenia
3
Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljna, Slovenia
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2020, 8(10), 1810; https://doi.org/10.3390/math8101810
Received: 7 September 2020 / Revised: 2 October 2020 / Accepted: 10 October 2020 / Published: 16 October 2020
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. View Full-Text
Keywords:
Show Figures

Figure 1

MDPI and ACS Style

Krnc, M.; Škrekovski, R. Group Degree Centrality and Centralization in Networks. Mathematics 2020, 8, 1810.