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). Q- linked to the bridging node x. It is, however, well measures are essentially a variant of betweenness known that information in a network may travel centrality that considers only geodesics between through several intermediary nodes. Hence, one can nodes from different groups. Assume that there are T also imagine indicators of brokerage that account for actors (nodes) in the network. Group G contains m situations where the bridging node is not necessarily nodes, while the other group, denoted as H, contains directly linked to the outer nodes. This is the case for n nodes; hence, T=m+n. If actor a belongs to group the gefura measures studied in this paper. G, and assuming for simplicity that actor a is gm, When it comes to brokerage, bridging, or gate- then the Q-measure for this actor, Q(a), is defined as keepership, Burt’s theory of social capital and struc- tural holes (Burt, 2004) should be mentioned. Struc- mn1 1 pagh, () tural holes in a social network are described as dis- Qa() ij . (3) (1)mn ij11 p connected or poorly connected areas between other- ghij, wise densely connected groups of people. Brokers play the role of connectors between two or more Here, p denotes the number of shortest paths gij,h poorly connected groups (Fig. 2) and are rewarded from the ith to the jth node, while pa() is the for this by an increase in social capital. Burt’s theory ghij, applies only to social networks and does not start number of shortest paths from the ith to the jth node from predefined groups; instead, groups are deter- that pass through node a. mined by the network structure. Using a slightly simpler notation, the same for- A third line of research pertaining to brokerage mula can be expressed as follows: was initiated by Flom et al. (2004) and later elabo- rated in work of the authors and colleagues. This 1 pagh, () research has introduced a ‘family’ of measures, Qa() , g h a . (4) (1)mn gG pgh, which are different from the work of Gould and hH 314 Guns et al. / Front Inform Technol Electron Eng 2015 16(4):311-320
Q-measures were applied in Rousseau (2005), brokerage is not optimal at all. Chen and Rousseau (2008), and Zhang et al. (2009). As these measures gauge the bridging role of Whereas the initial definition applied to connected, nodes, the term ‘gefura measure’ (after old Greek unweighted, and undirected networks, Rousseau and γεφυρα), meaning bridge measure, might be a more Zhang (2008) considered Q-measures in the context descriptive term with universal appeal. From now on of weighted directed networks and proposed that in the authors intend to use the symbol Γ (capital gamma: this case the definition should be based on flow be- the first letter of gefura) instead of Q. tweenness centrality (Freeman et al., 1991) rather than shortest-path betweenness centrality as defined in Eq. (1). 3 Unnormalized gefura measures in directed The main restriction up to this point was that networks the definition allowed for only two groups, even though many real-world networks have three or more Most of the definitions in previous studies of node groups. This restriction was removed by Guns gefura or Q-measures are normalized to the interval and Rousseau (2009), who generalized the Q- [0, 1]. This is, for instance, the case in Eq. (3), where measure definition to any finite number of groups the division by (m−1)n ensures a maximum value of and introduced the distinction between ‘global’ Q- one. Guns and Rousseau (2009) noted that “one may measures (brokerage between all different groups) imagine circumstances, e.g., real-world transporta- and ‘local’ Q-measures (brokerage between one’s tion networks, where normalization, leading to a rel- own group and the other groups). The new defini- ative measure, is not optimal. Then no division […] tions were applied to international co-authorship is performed. This yields an absolute measure of networks (Guns and Liu, 2010; Guns et al., 2011). ‘bridgeness’.” More specifically, normalization Rousseau et al. (2013; 2014) focused on mathemati- makes it possible to compare gefura measures across cal properties of Q-measures, including a convex networks of different sizes. If this is not the purpose, decomposition, relations with betweenness centrality, then normalizing brings no real advantage. We there- and a characterization of nodes with a normalized Q- fore start the discussion by considering gefura measure equal to one (the highest possible value). In measures without normalization, the most straight- Liu et al. (2013) the authors used a binary tree as a forward forms. model for a hierarchical structure. All nodes situated In practice, many networks are directed; i.e., at the same height were assumed to form a group. Q- their links have an inherent direction. Examples in- measure values were determined for each node in a clude citation networks, telephone networks (a calls binary tree of arbitrary height. b), and any linear order or any hierarchy. Guns and We note that networks for which node sets con- Rousseau (2009) wrote that “all formulae for Q- sist of disjoint subsets can be considered as a special measures—the original Q-measures for two groups type of multilayer networks, as described in Boccaletti as well as Q-measures for any finite number of et al. (2014). groups—can be applied to directed networks as A note on naming: Colleagues working in social well.” Nevertheless, gefura measures (or Q-measures) network theory and marketing observed that the term in directed networks have so far been studied explic- ‘Q-measure’ is non-descriptive and, moreover, is itly only by Rousseau and Zhang (2008), who pro- used in other contexts, such as Tobin’s Q in market- posed a definition for Γ-measures in directed, ing (Brainard and Tobin, 1968), Q-analysis as a weighted networks (although they pay more attention mathematical technique to study and analyze struc- to the question ‘how to deal with link weights’). This tures (Atkin, 1972), and Q-measure as an infor- section and the following ones therefore assume that mation retrieval metric in a graded relevance context we are working in the setting of directed networks, (Sakai, 2007). The best known Q in network theory unless explicitly stated otherwise. The undirected is probably Newman and Girvan (2004)’s Q denot- case can easily be derived by treating the undirected ing modularity in a network. So, indeed, the use of network as a symmetric directed network, in which the symbol Q and the term Q-measure to study each undirected link is replaced by two links Guns et al. / Front Inform Technol Electron Eng 2015 16(4):311-320 315
between the same node pair but pointing in opposite while the dashed line indicates an inward geodesic directions. from b3 to a3. Node a1 is part of the outward geodesic, If we have a network with three or more groups, but not of the inward one. This is an important dis- the basic procedure remains the same: the gefura tinction. For instance, in an author citation network measure of node a is calculated by determining the with countries as groups, it makes a profound differ- proportion of geodesics between nodes from differ- ence whether one forms an outward bridge (citing ent groups that pass through a. In this case, one authors from other countries) or an inward one (re- might pose the question exactly which groups are ceiving international citations). Hence, we propose taken into account. That is, one does not necessarily that one can distinguish between three cases: inward, need to consider geodesics between nodes from all outward, and both. We will denote these as ΓL(i), ΓL(o), groups but could restrict oneself to geodesics be- and ΓL(b), respectively. They are defined as follows: tween nodes from a subset of groups. While specific situations may warrant including or excluding specific pahg, () groups, we think that three situations are potentially L(i) ()a , (6) gG d phg, relevant to any network divided into more than two hG d groups (Rousseau et al., 2013): pagh, () 1. Global: all group pairs are taken into account. L(o) ()a , (7) gG d pgh, 2. Local: all pairs between a node’s own group hG d and other groups are taken into account. pagh,,() pa hg () 3. External: all group pairs, except those involv- ()a L(b) pp gG d gh,, hg (8) ing a node’s own group, are taken into account. hG d Before moving on to the definitions, we intro- L(i)()aa L(o) (). duce some notations and assumptions. Consider a network subdivided into S non-overlapping groups Here, node g always belongs to the same group as a. (1 The global gefura measure is then defined as follows: pa() ()a gh, . (5) G ghV, pgh, group(gh ) group( ) The only difference between this formula and unnormalized betweenness centrality is the re- Fig. 3 Inward and outward shortest paths in a directed striction that g and h belong to different groups. network For local gefura measures we essentially have There are two node groups, {a1, a2, a3} and {b1, b2, b3} one ‘special’ group (a node’s own group, here called Gd), and consider shortest paths between this group If there are only two groups (S=2), then ΓL(a)= and all the other groups. In a directed network we ΓG(a). can distinguish between outward geodesics that orig- Finally, the external gefura measure is defined inate from Gd and end in another group and inward as geodesics that originate from another group and end pagh, () in Gd (Fig. 3). If a1 is the node under study, the thick ()a . (9) E p line indicates an outward geodesic from a3 to b3, jkdgGhG jk, gh, 316 Guns et al. / Front Inform Technol Electron Eng 2015 16(4):311-320 Note that the first summation considers all pos- Likewise, one can define a structural local ge- sible j and k (both different from d). Hence, a given fura measure. Again, we distinguish between inward, group once plays the role of Gj and once the role of outward, and both: Gk. If S=2, the external gefura measure is equal to zero. S 11pahg, () One can easily see that there is no overlap be- ()a , (12) L(i) SP1 p tween the group pairs on which Γ and Γ are lddl,, g Gd hg L(b) E hG l based. Furthermore, the group pairs for Γ are a dis- G joint union of those used to calculate Γ and Γ . 11pa() L(b) E S ()a gh, , (13) More specifically, the relationship between global, L(o) SP1 p lddl,, g Gd gh local, and external gefura measures is as follows: hG l S 11pagh,,() pa hg () ΓG=ΓL(b)+ΓE. (10) ()a L(b) 2(SP 1) pp lddl,,, g Gd gh hg hG l 4 Normalized gefura measures in directed SS()aa () networks L(i) L(o) . (14) 2 The next step is to define the same measures, global, local, and external gefura, with normalization. In the case of an undirected network, we have Depending on the relative importance one gives to (Guns and Rousseau, 2009) groups or nodes we come up with two definitions. S 11pagh, () 4.1 Structural normalization ()a L 2(SPp 1) lddl,, g Gd gh The structural global gefura measure, denoted hG l as S , is defined as follows (Guns and Rousseau, 1 G dl,, ld (15) 2(S 1) 2009): ld 1 dl, . S 1 ld 11pa() S ()a gh, G SS(1) P p kl, kl,, g Gk gh Also, note that here the global and local hG (11) l measures coincide when S=2. Indeed, 1 kl, ().a SS(1) kl, 2 1 G,,()aaa dl () ld () 2(2 1) The symbol Pk,l refers to the number of possible 2 2 combinations of elements belonging to groups G dl,L()aa (). k 2 and Gl. Hence, Pk,l=|Gk\{a}|·|Gl\{a}|, where |·| refers to the number of elements in the set. The part be- Finally, we define the structural external gefura tween |·| is essentially a normalized gefura measure S measure (Rousseau et al., 2013): for two groups G and G (see Eq. (4)); this can be E k l called a partial gefura measure and is denoted as Γk,l S 1 (note that, in general, Γk,l(a)≠Γl,k(a)). Two normaliza- E,()aa kl (). (16) (1)(2)SSkl, tions are applied here: first, an ‘inner’ normalization kld in the partial gefura measure, and second, an ‘outer’ normalization for the number of groups. In the spe- Rousseau et al. (2013; 2014) proved the follow- cial case Pk,l=0, which occurs if k or l equals d and ing convex decomposition of the global gefura Gd={a}, we apply the rule 0/0=0. measure: Guns et al. / Front Inform Technol Electron Eng 2015 16(4):311-320 317 22S In all these cases it might be necessary to apply SS()aa () S (). a (17) GLSS E the rule 0/0=0. The basic external gefura measure B is defined as This relation still holds for a directed network. G 4.2 Basic normalization B 1 pagh, () E ()a , (23) We have seen how structural normalization in- Mpagh,0gG k , hG l volves an ‘inner’ and an ‘outer’ normalization. An kld alternative starts from the unnormalized definitions and uses just one normalization step. As this procedure is where Ma,0=∑k≠l≠dmkml. Note that each product of the somewhat simpler, we call it basic normalization. form mkml occurs also as mlmk in this sum. B The basic global gefura measure G is defined Similar to the structural gefura measures, a con- as follows: vex decomposition of the global measure can be obtained: pa() B 1 gh, G ()a , (18) BBB MpgG aghk , GLE()aX () aYa (), (24) hG l kl where where the symbol Ma is defined as (1)()mNm X dd, (25) M ak|\{}||\{}|.Ga Ga l (1)()mNmmmddkl kl, kl, kld Note that this is simply the unnormalized global mm kl M gefura measure (5) divided by a maximum, to obtain Y kld a,0 . (26) a value between zero and one. (1)()mNmmmMddkla kl, The basic normalization of local gefura measures kld also applies to the inward, outward, and both cases: 1 pahg, () 4.3 Comparison of structural and basic gefura B ()a , (19) L(i) measures (1)()mNmpddhggG d , hG d The difference between structural and basic ge- 1 pa() B ()a gh, , (20) fura measures can be clarified by rewriting the for- L(o) (1)()mNmp ddghgG d , mer to be more similar to the latter. We will use the hG d structural global gefura measure here, but a similar 1 papa() () B ()a hg,, gh reasoning applies to the local and external measures. L(b) 2(mNm 1)( ) p p ddgG d hggh,, For simplicity, we will discuss this within the setting hG d of an undirected network, but the reasoning is similar BB()aa () L(i) L(o) . (21) for directed networks. Eq. (11) can be written as 2 follows: In the case of undirected networks, this can be simplified to S 1 pagh, () G,,()aw jka , (27) Mpaghjk,, g Gjk h G , 1 pa() pa () jk B ()a hg,, gh L 2(mNm 1)( ) p p ddhgghgG d ,,where hG d 1 pa() M 1 gh, . (22) w a . (28) jka,, SS(1)|\{}||\{}| G a G a (1)()mNmpddghgG d , jk hG d 318 Guns et al. / Front Inform Technol Electron Eng 2015 16(4):311-320 Whereas the basic normalization or the unnor- In conclusion, structural normalization attaches malized form treats all geodesics the same, the above more importance to the group level (answering the equations show that structural normalization is a sort question: for how many group pairs is the node a of weighted form of basic normalization. More spe- broker?), whereas basic normalization attaches more cifically, the ratio pg,h(a)/pg,h is multiplied with a fac- importance to the node level (answering the question: tor wj,k,a, such that more weight is given to geodesics for how many pairs of nodes from different groups is between smaller groups. The result is that structural the node a broker?). normalization treats a small group as equal to a large one, whereas basic normalization gives equal weight to each geodesic regardless of the group size. a2 A few examples may help to clarify this differ- ence. Consider example network A in Fig. 4, where a1 nodes are grouped according to their initial letter: a1 and a2 form one group; b1, b2, and b3 form another; d2 b2 and so on. It can easily be seen that only nodes a1, b1, b1 SS c1 and c1 have Γ>0. We find that G1()ab G1 () S ()c 2/3, whereas BB()ab 0.37 () G1 G1 G1 d1 b3 B 0.6G1 (c ) 0.71. The values of the structural ge- c2 fura measure are the same, because the three nodes bridge between the same number of groups. However, Fig. 5 Example network B c1’s basic gefura measure is the highest because this node bridges between more node pairs than the other two, and likewise, b1 obtains a higher basic gefura 5 An adapation of the Brandes algorithm value than a1. This illustrates how structural normal- ization normalizes at the group level, while basic Brandes (2001) introduced a fast algorithm to normalization works at the node level. determine betweenness centrality. It works in O(nm) time for unweighted networks and O(n+m) space, where n denotes the number of nodes and m the number of links. In a follow-up article, Brandes (2008) showed how the basic algorithm can be adapted to calculate many variants of betweenness centrality, including gefura measures for two groups (with the same asymptotic time complexity). Essen- tially the same algorithm can be used to calculate (unnormalized or basic) global gefura measures. Structural gefura measures cannot be determined this way and we currently lack an efficient algorithm to determine them. This is a practical disadvantage of Fig. 4 Example network A structural gefura. We reuse the notation of Brandes (2008). The algorithm considers each node once; this node is Using the same underlying network but a dif- referred to as the ‘source’. Here, S is a stack of nodes, ferent group partitioning (Fig. 5), we have S ()c G1 Pred[v] is a list of predecessors on the shortest path SS BBB G1()ba G () 1 and G1()abc G1 () G1 (). The from the source to vV, σ[v] is the number of geo- difference is that c1 is now also a bridge between the desics from the source to v, and δ[v] is the dependency a- and d-group, and between the b- and d-group. of the source on v. That is, if we denote the source by Guns et al. / Front Inform Technol Electron Eng 2015 16(4):311-320 319 g, then []vpvp ( ()/ ). The crucial ac- adapted to this structure used to be called Q- hV g,,hgh measures, but we explained why this was not a good cumulation step then becomes: terminological choice, preferring the name ‘gefura measures’ instead. These measures were designed as Accumulation: { indicators of a node’s brokerage role between such for vV do δ[v]←0 groups. We started by a systematic study of unnor- while S not empty do malized gefura measures in directed networks and pop w←S showed that, next to the ‘structural’ normalization if group(s)≠group(w) then i←1 else i←0 that has hitherto been applied, a ‘basic’ normaliza- for vPred[w] do tion procedure is possible. While the former normal- []v []vv [] iw [ ] izes at the level of groups, the latter normalizes at the []w level of nodes. The undirected case follows straight- if w≠s then Γ (w)←Γ (w)+δ[w] G G forwardly by treating undirected networks as sym- } metric directed networks. Finally, Brandes’ algo- rithm for betweenness centrality was adjusted to With a small adaptation, the algorithm can be used as cover most of these cases. This led to the open prob- follows to determine local gefura: lem of adapting this algorithm to calculate gefura measures in their structural normalization form. We Accumulation: { observed that the above-mentioned partition refers to for vV do δ[v]←0 nodes. Links may connect nodes belonging to the while S not empty do same group or may connect nodes belonging to dif- pop w←S ferent nodes. if group(s)≠group(w) then i←1 else i←0 We are convinced that gefura measures consti- for vPred[w] do tute a useful tool for social scientists, informetricians, []v []vv [] iw [ ] and colleagues studying complex networks, in the []w case the studied networks are partitioned into if w≠s and group(s)=group(w) then subgroups. ΓL(w)←ΓL(w)+δ[w] } References Atkin, R.H., 1972. From cohomology in physics to q- The above algorithms apply to both undirected connectivity in social science. Int. J. Man-Mach. Stud., 4(2):139-167. [doi:10.1016/S0020-7373(72)80029-4] and directed networks. Here, undirected networks are Barrat, A., Barthélemy, M., Pastor-Satorras, R., et al., 2004. treated like symmetric directed networks as well. The architecture of complex weighted networks. PNAS, The algorithm for local gefura yields the inward lo- 101(11):3747-3752. [doi:10.1073/pnas.0400087101] cal gefura measure when applied to a directed net- Boccaletti, S., Bianconi, G., Criado, R., et al., 2014. The work. One can obtain the outward local gefura by structure and dynamics of multilayer networks. Phys. reversing the direction of all links prior to using the Rep., 544(1):1-122. [doi:10.1016/j.physrep.2014.07.001] Brainard, W.C., Tobin, J., 1968. Econometric models: their algorithm. problems and usefulness. Pitfalls in financial model An implementation of this algorithm in Python building. Amer. Econ. Rev., 58(2):99-122. can be found at https://github.com/rafguns/gefura. Brandes, U., 2001. A faster algorithm for betweenness cen- trality. J. Math. Sociol., 25(2):163-177. [doi:10.1080/ 0022250X.2001.9990249] 6 Conclusions Brandes, U., 2008. On variants of shortest-path betweenness centrality and their generic computation. Soc. Netw., 30(2):136-145. [doi:10.1016/j.socnet.2007.11.001] In this contribution we studied partitioned net- Burt, R.S., 2004. 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