Guns et al. / Front Inform Technol Electron Eng 2015 16(4):311-320 311 Frontiers of Information Technology & Electronic Engineering www.zju.edu.cn/jzus; engineering.cae.cn; www.springerlink.com ISSN 2095-9184 (print); ISSN 2095-9230 (online) E-mail: [email protected] Unnormalized and normalized forms of gefura measures in * directed and undirected networks Raf GUNS1, Ronald ROUSSEAU‡1,2 (1Institute for Education and Information Sciences, University of Antwerp, Venusstraat 35, Antwerp B-2000, Belgium) (2Department of Mathematics, KU Leuven, Celestijnenlaan 200B, Leuven B-3001, Belgium) E-mail: [email protected]; [email protected]; [email protected] Received Dec. 9, 2014; Revision accepted Mar. 12, 2015; Crosschecked Mar. 13, 2015 Abstract: In some networks nodes belong to predefined groups (e.g., authors belong to institutions). Common network cen- trality 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 normal- izes at the level of nodes. (2) Treating undirected networks as equivalent to symmetric directed networks, we expand the defini- tion of gefura measures to the directed case. (3) It is shown how Brandes’ algorithm for betweenness centrality can be adjusted to cover these cases. Key words: Networks subdivided in groups, Partitions, Gefura measures, Q-measures, Brokerage role, Directed and undirected networks, Brandes’ algorithm doi:10.1631/FITEE.1400425 Document code: A CLC number: TP393; G350 1 Introduction are nowadays studied from a network perspective. Large-scale analyses of so-called complex networks Networks are abundant. Indeed, road maps rep- reveal that the same structural features, such as resent the network of cities and highways, and simi- skewed degree distributions and local clustering, can larly we have other transportation networks such as emerge in different fields (Christensen and Albert, the worldwide air transport network, metabolic net- 2007). This underlines the importance of network works (Barrat et al., 2004; Guimerà and Amaral, studies. 2005; Guimerà et al., 2005), and the shipping and The field of informetrics is no exception to this harbor network. The network that is probably best trend (e.g., Otte and Rousseau (2002) and Ding known, the Internet, consists of a worldwide assem- (2011)). Maps of science are constructed based on blage of local, regional, and global academic, busi- the complete Web of Knowledge, Scopus, etc. Top- ness, government, private, and public computer net- ics such as collaboration, diffusion, and citation have works. Many research topics across all disciplines been studied frequently from the perspective of so- cial network analysis. Moreover, a whole new sub- ‡ Corresponding author field related to the Internet and the World Wide Web, * Project supported by the National Natural Science Foundation of namely webmetrics, has emerged within informetrics. China (No. 71173154) In some networks, nodes belong to predefined ORCID: Raf GUNS, http://orcid.org/0000-0003-3129-0330; Ronald ROUSSEAU, http://orcid.org/0000-0002-3252-2538 groups. For instance, a network of friendships in © Zhejiang University and Springer-Verlag Berlin Heidelberg 2015 school may have pupils as nodes and classes as 312 Guns et al. / Front Inform Technol Electron Eng 2015 16(4):311-320 groups. Likewise, a citation network may have arti- and a set of links, arcs, or edges (E). Each link (u, v) cles as nodes and journals as groups. Previous re- E is a connection from u to v (u, vV). The number search has introduced so-called Q-measures (Flom et of shortest paths or geodesics from node g to h (in al., 2004), or the newer and preferred term, gefura that order) is denoted as pg,h. The number of geodes- measures, as indicators of a node’s brokerage role ics from g to h that pass through a node a (a≠g, a≠h, between groups. In this article we make the following so a is not an endpoint) is denoted as pg,h(a). Of new contributions to the study on gefura measures: course, in an undirected network, pg,h=ph,g, and simi- 1. Analogous to the treatment of betweenness larly pg,h(a)=ph,g(a), while this is usually not the case centrality by Brandes (2001; 2008), we start the dis- in a directed network. cussion by considering gefura measures in directed Networks can be characterized by several dif- networks. Undirected networks are then equivalent ferent measures. Centrality measures are indicators to symmetric directed networks. That is, each undi- that characterize the importance of individual nodes. rected link {a, b} is equivalent to two directed links The most important ones are degree centrality, (a, b) and (b, a). Fig. 1 contains an example. closeness centrality, betweenness centrality, and ei- genvector centrality, referred to as rank prestige by Wasserman and Faust (1994). We focus on between- ness centrality because of its importance to the fol- lowing discussion. Betweenness centrality character- izes a node’s control over the geodesic information Fig. 1 Undirected network (a) treated as equivalent to a flow through the network. The betweenness centrality directed network with bidirectional links (b) of node a, denoted as CB(a), is defined as 2. We systematically study unnormalized gefura pagh, () Ca() , (1) measures and show that, next to the ‘structural’ nor- B gh, V pgh, malization that has hitherto been applied, a ‘basic’ normalization procedure is possible. where we assume that g, h, and a are three different 3. We show that ‘structural’ normalization pays nodes. From now on we will use the notation g≠h≠a, more attention to the group level, whereas ‘basic’ nor- which should be understood as g≠h, g≠a, and h≠a. malization pays more attention to the level of nodes. By convention, we set 0/0=0. In this way, the formula 4. Building on the work of Brandes (2008), an of betweenness centrality and subsequent formulae efficient algorithm is introduced to calculate unnor- in this paper can also be applied to unconnected net- malized or basic gefura measures in both directed works (Freeman, 1977). This convention is equiva- and undirected networks. lent to considering only those node pairs (g, h) where The main aim of this paper is to provide a gen- h is reachable from g. eral theoretical framework for studying the bridging Since we treat undirected networks as symmet- role of nodes in networks with predefined groups. ric directed networks, each path in an undirected This article is an extensive elaboration of some ideas network is counted twice. For instance, one would presented in Rousseau et al. (2015). We apply a normally claim that the undirected network in Fig. 1 slightly other notation than in our previous articles contains one geodesic between b and d (b-c-d), on this topic. This new notation corresponds better whereas the symmetric directed interpretation yields with the case of directed networks. two geodesics (b-c-d and d-c-b). Hence, when apply- ing Eq. (1) to an undirected network, one should di- vide the outcome by two. 2 Background The maximum value of CB(a) in Eq. (1) de- 2.1 Networks and centrality pends on the size of the network. If one wants to compare values between nodes from different net- We assume that we have a directed network works, normalization to values between 0 and 1 can =(V, E), consisting of a set of nodes or vertices (V) be applied. For a network with N nodes, this becomes Guns et al. / Front Inform Technol Electron Eng 2015 16(4):311-320 313 Fernandez and/or Burt in several ways: N 1 pagh, () CaB () . (2) 1. No assumptions on the type of network need (1)(2)NNghV, p gh, to be made (connected or disconnected, directed or undirected). The highest value reached by normalized be- 2. The measures are agnostic to setting; i.e., tweenness centrality is by definition the value one. It they can also be applied outside a social network is obtained by the center of a star network. context. 2.2 Brokerage between groups in a network 3. The measures quantify a node’s bridging role even if it is not directly linked to the outer nodes. In some cases, a network’s nodes may belong to 4. Groups are well-defined from the outset and different groups. We assume that node groups are brokers belong, in principle, to one of these groups known (which clearly separates our work from, for (although we do not exclude the case that a broker is instance, research on community finding algorithms). a singleton group on its own). Brokerage can informally be understood as the ex- tent to which a node facilitates information exchange between other nodes, especially nodes that belong to different groups. Brokerage between disjoint groups in networks has previously been studied by Gould and Fernandez (1989), who studied subnetworks of the form a-x-b, where a, x, and b are nodes. Depend- Fig. 2 Example of brokerage according to Burt ing on the question whether or not these nodes be- long to the same or different groups, they distin- Flom et al. (2004) began this line of research by guished between five different brokerage roles for x. introducing ‘Q-measures’ as an indicator of broker- Gould and Fernandez (1989) considered only age in an undirected network where nodes belong to situations where the ‘outer’ nodes a and b are directly one of two groups (e.g., males and females).