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Group Degree Centrality and Centralization in Networks

Author

Listed:
  • Matjaž Krnc

    (FAMNIT, University of Primorska, 6000 Koper, Slovenia
    These authors contributed equally to this work.)

  • Riste Škrekovski

    (FAMNIT, University of Primorska, 6000 Koper, Slovenia
    Faculty of Information Studies, 8000 Novo Mesto, Slovenia
    Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljna, Slovenia
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Matjaž Krnc & Riste Škrekovski, 2020. "Group Degree Centrality and Centralization in Networks," Mathematics, MDPI, vol. 8(10), pages 1-11, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1810-:d:429165
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    References listed on IDEAS

    as
    1. Bell, Jocelyn R., 2014. "Subgroup centrality measures," Network Science, Cambridge University Press, vol. 2(2), pages 277-297, August.
    2. Dorit S. Hochbaum & Anu Pathria, 1998. "Analysis of the greedy approach in problems of maximum k‐coverage," Naval Research Logistics (NRL), John Wiley & Sons, vol. 45(6), pages 615-627, September.
    3. Stephen P. Borgatti, 2006. "Identifying sets of key players in a social network," Computational and Mathematical Organization Theory, Springer, vol. 12(1), pages 21-34, April.
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