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| Unnormalized and normalized forms of gefura measures in directed and undirected networks |
| Raf Guns, Ronald Rousseau |
| Institute for Education and Information Sciences, University of Antwerp, Venusstraat 35, Antwerp B-2000, Belgium; Department of Mathematics, KU Leuven, Celestijnenlaan 200B, Leuven B-3001, Belgium |
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Abstract In some networks nodes belong to predefined groups (e.g., authors belong to institutions). Common network centrality measures do not take this structure into account. Gefura measures are designed as indicators of a node’s brokerage role between such groups. They are defined as variants of betweenness centrality and consider to what extent a node belongs to shortest paths between nodes from different groups. In this article we make the following new contributions to their study: (1) We systematically study unnormalized gefura measures and show that, next to the ‘structural’ normalization that has hitherto been applied, a ‘basic’ normalization procedure is possible. While the former normalizes at the level of groups, the latter normalizes at the level of nodes. (2) Treating undirected networks as equivalent to symmetric directed networks, we expand the definition of gefura measures to the directed case. (3) It is shown how Brandes’ algorithm for betweenness centrality can be adjusted to cover these cases.
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Received: 09 December 2014
Published: 03 April 2015
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有向网络和无向网络中桥接（gefura）测度的非归一化和归一化形式
目的：针对现实生活中某些网络节点属于预先定义的小组而常规网络其中心性测度并未考虑到这一结构特点，本文以桥接测度作为小组间中介节点的指标并研究之。
创新点：系统性研究非归一化桥接测度，提出除“结构化”归一化（组级别）之外“基本”归一化（节点级别）也是可行的。将无向网络视为对称有向网络，将桥接测度定义推广至有向网络。
方法：首先系统性研究有向网络中的非归一化桥接测度。然后研究有向网络中的归一化桥接测度，指出除“结构化”归一化（组级别）之外，“基本”归一化（节点级别）也是可行的。将无向网络视为对称有向网络，所得结论对于无向网络同样成立。最后，说明如何调节Brandes算法使之应用于测量所提网络的中介中心性。
结论：对于研究复杂网络的社会学家、信息计量学者等，若其所研究的网络被分割为小组，则桥接测度将是十分有用的工具。
关键词：
细分为组的网络,
分割,
桥接（gefura）测度,
Q-测度,
中介角色,
有向网络和无向网络,
Brandes算法
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