Unnormalized and normalized forms of gefura measures in directed and undirected networks*

Guns Raf; Rousseau Ronald

doi:10.1631/FITEE.1400425

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Unnormalized and normalized forms of gefura measures in directed and undirected networks*

Computer & Automation | Updated：2022-06-14

- Unnormalized and normalized forms of gefura measures in directed and undirected networks*
- 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
- Frontiers of Information Technology & Electronic Engineering Vol. 16, Issue 4, Pages: 311-320(2015)
- Affiliations：
  
  1.Institute for Education and Information Sciences, University of Antwerp, Venusstraat 35, Antwerp B-2000, Belgium
  2.Department of Mathematics, KU Leuven, Celestijnenlaan 200B, Leuven B-3001, Belgium
- Author bio：
  
  ‡Corresponding Author
- Funds：
- DOI：10.1631/FITEE.1400425
  CLC：
- Received：09 December 2014，
  
  Revised：2015-03-13，
  
  Accepted：12 March 2015，
  
  Published：2015-04
- Accepted：
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Guns Raf, Rousseau Ronald. Unnormalized and normalized forms of gefura measures in directed and undirected networks*[J]. Frontiers of Information Technology & Electronic Engineering, 2015, 16(4): 311-320.
DOI：

Guns Raf, Rousseau Ronald. Unnormalized and normalized forms of gefura measures in directed and undirected networks*[J]. Frontiers of Information Technology & Electronic Engineering, 2015, 16(4): 311-320. DOI： 10.1631/FITEE.1400425.

摘要

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。

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.

关键词

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references

RH Atkin . From cohomology in physics to q -connectivity in social science . Int J Man-Mach Stud , 1972 . 4 ( 2 ): 139 - 167 . DOI: 10.1016/S0020-7373(72)80029-4 http://doi.org/10.1016/S0020-7373(72)80029-4 .

A Barrat , M Barthélemy , R Pastor-Satorras , 等 . The architecture of complex weighted networks . PNAS , 2004 . 101 ( 11 ): 3747 - 3752 . DOI: 10.1073/pnas.0400087101 http://doi.org/10.1073/pnas.0400087101 .

S Boccaletti , G Bianconi , R Criado , 等 . The structure and dynamics of multilayer networks . Phys Rep , 2014 . 544 ( 1 ): 1 - 122 . DOI: 10.1016/j.physrep.2014.07.001 http://doi.org/10.1016/j.physrep.2014.07.001 .

WC Brainard , J Tobin . Econometric models: their problems and usefulness. Pitfalls in financial model building . Amer Econ Rev , 1968 . 58 ( 2 ): 99 - 122 . .

U Brandes . A faster algorithm for betweenness centrality . J Math Sociol , 2001 . 25 ( 2 ): 163 - 177 . DOI: 10.1080/0022250X.2001.9990249 http://doi.org/10.1080/0022250X.2001.9990249 .

U Brandes . On variants of shortest-path betweenness centrality and their generic computation . Soc Netw , 2008 . 30 ( 2 ): 136 - 145 . DOI: 10.1016/j.socnet.2007.11.001 http://doi.org/10.1016/j.socnet.2007.11.001 .

RS Burt . Structural holes and good ideas . Amer J Sociol , 2004 . 110 ( 2 ): 349 - 396 . DOI: 10.1086/421787 http://doi.org/10.1086/421787 .

LX Chen , R Rousseau . Q-measures for binary divided networks: bridges between German and English institutes in publications of the J. Fluid Mech . Scientometr , 2008 . 74 ( 1 ): 57 - 69 . DOI: 10.1007/s11192-008-0103-6 http://doi.org/10.1007/s11192-008-0103-6 .

C Christensen , R Albert . Using graph concepts to understand the organization of complex systems . Int J Bifurc Chaos , 2007 . 17 ( 7 ): 2201 - 2214 . DOI: 10.1142/S021812740701835X http://doi.org/10.1142/S021812740701835X .

Y Ding . Scientific collaboration and endorsement: network analysis of coauthorship and citation networks . J Inform , 2011 . 5 ( 1 ): 187 - 203 . DOI: 10.1016/j.joi.2010.10.008 http://doi.org/10.1016/j.joi.2010.10.008 .

PL Flom , SR Friedman , S Strauss , 等 . A new measure of linkage between two sub-networks . Connections , 2004 . 26 ( 1 ): 62 - 70 . .

LC Freeman . A set of measures of centrality based on betweenness . Sociometry , 1977 . 40 ( 1 ): 35 - 41 . .

LC Freeman , SP Borgatti , DR White . Centrality in valued graphs: a measure of betweenness based on network flow . Soc Netw , 1991 . 13 ( 2 ): 141 - 154 . DOI: 10.1016/0378-8733(91)90017-N http://doi.org/10.1016/0378-8733(91)90017-N .

RV Gould , RM Fernandez . Structures of mediation: a formal approach to brokerage in transaction networks . Sociol Method , 1989 . 19 89 - 126 . DOI: 10.2307/270949 http://doi.org/10.2307/270949 .

R Guimerà , LAN Amaral . Functional cartography of complex metabolic networks . Nature , 2005 . 433 895 - 900 . DOI: 10.1038/nature03288 http://doi.org/10.1038/nature03288 .

R Guimerà , S Mossa , A Turtschi , 等 . The worldwide air transportation network: anomalous centrality, community structure, and cities’ global roles . PNAS , 2005 . 102 ( 22 ): 7794 - 7799 . DOI: 10.1073/pnas.0407994102 http://doi.org/10.1073/pnas.0407994102 .

R Guns , YX Liu . Scientometric research in China in context of international collaboration . 2010 . Proc. 6th Int. Conf. on Scientometrics and University Evaluation . 112 - 115 . .

R Guns , R Rousseau . Gauging the bridging function of nodes in a network: Q-measures for networks with a finite number of subgroups . 2009 . Proc. 12th ISSI . 131 - 142 . .

R Guns , YX Liu , D Mahbuba . Q-measures and betweenness centrality in a collaboration network: a case study of the field of informetrics . Scientometrics , 2011 . 87 ( 1 ): 133 - 147 . DOI: 10.1007/s11192-010-0332-3 http://doi.org/10.1007/s11192-010-0332-3 .

YX Liu , R Guns , R Rousseau . A binary tree as a basic model for studying hierarchies using Q-measures . SRELS J Inform Manag , 2013 . 50 ( 5 ): 521 - 528 . .

MEJ Newman , M Girvan . Finding and evaluating community structure in networks . Phys Rev E , 2004 . 69 026113.1 - 02611315 . DOI: 10.1103/PhysRevE.69.026113 http://doi.org/10.1103/PhysRevE.69.026113 .

E Otte , R Rousseau . Social network analysis: a powerful strategy, also for the information sciences . J Inform Sci , 2002 . 28 ( 6 ): 441 - 453 . DOI: 10.1177/016555150202800601 http://doi.org/10.1177/016555150202800601 .

R Rousseau . Q-measures for binary divided networks: an investigation within the field of informetrics . 2005 . Proc. Amer. Soc. Inform. Sci. Technol . ( 1 ): 675 - 696 . .

R Rousseau , L Zhang . Betweenness centrality and Q-measures in directed valued networks . Scientometrics , 2008 . 75 ( 3 ): 575 - 590 . DOI: 10.1007/s11192-007-1772-2 http://doi.org/10.1007/s11192-007-1772-2 .

R Rousseau , YX Liu , R Guns . Mathematical properties of Q-measures . J Inform , 2013 . 7 ( 3 ): 737 - 745 . DOI: 10.1016/j.joi.2013.06.002 http://doi.org/10.1016/j.joi.2013.06.002 .

R Rousseau , YX Liu , R Guns . An addendum and correction to “Mathematical properties of Q-measures” (vol. 7, issue 3, pp.737-745) . J Inform , 2014 . 8 ( 3 ): 486 - 490 . DOI: 10.1016/j.joi.2014.01.004 http://doi.org/10.1016/j.joi.2014.01.004 .

R Rousseau , R Guns , YX Liu . Gauging the bridging function of nodes in a network: the gefura measure . 2015 . Proc. 8th Int. Conf. on Scientometrics and University Evaluation, in press . .

T Sakai . On the reliability of information retrieval metrics based on graded relevance . Inform Process Manag , 2007 . 43 ( 2 ): 531 - 548 . DOI: 10.1016/j.ipm.2006.07.020 http://doi.org/10.1016/j.ipm.2006.07.020 .

S Wasserman , K Faust . Social Network Analysis: Methods and Applications , 1994 . Cambridge University Press, UK , .

WL Zhang , LC Yin , J Pang . The application of Q-measure to gender study in cooperation network . Sci Technol Progr Pol , 2009 . 26 ( 15 ): 100 - 103 . (in Chinese) .

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