- Email: [email protected]
A general conceptual framework for characterizing the ego in a network
Journal of Informetrics 9 (2015) 145–149
Contents lists available at ScienceDirect
Journal of Informetrics journal homepage: www.elsevier.com/locate/joi
A general conceptual framework for characterizing the ego in a network Ronald Rousseau a,b , Star X. Zhao c,∗ a b c
Universiteit Antwerpen (UA), IBW, Stadscampus, Venusstraat 35, 2000 Antwerpen, Belgium KU Leuven, Department of Mathematics, 3000 Leuven, Belgium Department of Information Science, Business School, East China Normal University, 200241 Shanghai, China
a r t i c l e
i n f o
Article history: Received 13 September 2014 Received in revised form 7 December 2014 Accepted 8 December 2014 Keywords: Network analysis h-Index h-Degree a-Index g-Index Zipf list
a b s t r a c t In this contribution we consider one particular node in a network, referred to as the ego. We combine Zipf lists and ego measures to put forward a conceptual framework for characterizing this particular node. In this framework we unify different forms of h-indices, in particular the h-degree, introduced in the literature. Similarly, different forms of the gindex, the a-index and the R-index are unified. We focus on the pure mathematical and logical concepts, referring to the existing literature for practical examples. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction An interesting recent development in network analysis is the emergence of h-type network measures such as the lobby index (Korn, Schubert, & Telcs, 2009), the h-degree (Zhao, Rousseau, & Ye, 2011), the partnership ability index (Schubert, 2012a), the Hw-degree (Abbasi, 2013) and the C-index (Yan, Zhai, & Fan, 2013). These indicators provide new approaches for exploring networks. Moreover, empirical investigations (Schubert, 2012a; Zhao, Zhang, Li, Tan, & Ye, 2014) show that the Schubert-Glänzel model (2007) for the h-index is applicable in networks. In this contribution we focus on one particular node in a network, referred to as the ego. As such our approach differs from e.g. (Glänzel, 2012) where the main focus is on the network as a whole. We combine Zipf lists, defined further on, and ego measures to put forward a conceptual framework for characterizing this particular node in networks. In this framework, also the original h-index can be converted into an h-degree. In this way we unify approaches introduced by the authors mentioned above and offer a birds-eye-view on the topic of h-type indicators as applied to networks. Focus is on the pure mathematical and logical concepts, referring the reader for practical examples to the existing literature.
∗ Corresponding author at: Room 437, Building of Law & Business, 500 Dongchuan Rd. (Minhang Campus of East China Normal University), 200241 Shanghai, China. Tel.: +86 15800587558. E-mail addresses: [email protected], [email protected] (R. Rousseau), [email protected] (S.X. Zhao). http://dx.doi.org/10.1016/j.joi.2014.12.002 1751-1577/© 2014 Elsevier Ltd. All rights reserved.
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2. The ego in an undirected, unweighted network In this section we consider an undirected, unweighted network and consider different ways to characterize the ego, node n, as a function of its immediate neighbourhood. Well-known centrality measures such as closeness centrality, eccentricity, betweenness centrality, Katz’ influence and eigenvector centrality are not considered as for their calculation (extremely small networks being an exception) one needs information beyond a node’s immediate neighbourhood (Bonacich, 1987; Katz, 1953; Otte & Rousseau, 2002; Wasserman & Faust, 1994). The first-order neighbourhood of node n is denoted as N1 (n) and consists of all nodes linked to n. Such nodes are referred to as first-order neighbours. The second-order neighbourhood of n consists of all nodes linked to a first order neighbour, node n and all first-order neighbours being excluded. This set is denoted as N2 (n). When it is clear that we are only using first-order neighbours, we denote the first-order neighbourhood of node n as N(n). 2.1. A binary measure The binary measure c(n) takes the value 1 if node n has at least one neighbour (hence one link) and the value 0 if it has not. This binary measure distinguishes between unconnected singletons and nodes that are connected to a least one other node. 2.2. Degree centrality (Wasserman & Faust, 1994) Degree centrality of node n, denoted as deg(n), is equal to the number of nodes in node n’s first-order neighbourhood; hence it is equal to the number of links adjacent to n. 2.3. The lobby index Schubert, Korn, and Telcs (2009) defined the lobby index of node n in an undirected unweighted network as the largest integer l(n) such that node n has at least l(n) neighbours with at least degree centrality l(n). 2.4. The second-order lobby index The second-order lobby index is defined in a similar way as the (first-order) lobby index. The only difference is that one uses N2 (n) instead of N1 (n). Clearly it is possible to define k-th order lobby indices, but as we do not see any application of higher order lobby indices we do not go into this. 3. The ego in an undirected, weighted network In a weighted network a weight is associated with each link. For simplicity we assume that this weight is a natural number, different from zero (for now). 3.1. Node strength The node strength of a node in an undirected weighted network is defined as the sum of the strengths (or weights) of all its links (Barrat, Barthélemy, Pastor-Satorras, & Vespignani, 2004). This is denoted as dS (n). When all weights are equal to 1 (corresponding to the unweighted case) the node strength becomes the degree centrality of the node. 3.2. A node’s h-degree Zhao et al. (2011) introduced the h-degree of a node in a weighted undirected network as follows. Definition. The h-degree (dh ) of node n in a undirected weighted network is equal to dh (n), if dh (n) is the largest natural number such that n has dh (n) links each with strength at least equal to dh (n). If all weights are one or zero (zero, only when there is no link) then the h-degree coincides with the binary measure c(n). In 2012 Schubert proposed the partnership ability index, denoted as (Schubert, 2012a). This notion was introduced in a co-authorship network and defined as the largest natural number P such that an actor has at least P partners with whom he/she had at least P joint articles. It was pointed out that this definition could be adapted to other networks such as movie actor networks, sexual encounter networks, and cooperation between jazz musicians (Schubert, 2012a, 2012b). It is clear that all these cases are just illustrations of the h-degree in an undirected network (Rousseau, 2012; Schubert, 2012a). Indeed, it suffices using the number of joint articles, joint movies, sexual encounters and joint recording sessions and records as weights.
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3.3. The w-lobby index (Zhao et al., 2011) This definition extends the lobby index to the case of a weighted undirected network: the w-lobby index (weighted network lobby index) of node n, denoted as lw (n) is defined as the largest integer k such that node n has at least k neighbours with node strength at least k. Clearly, if links are unweighted, then the w-lobby index coincides with the original lobby index. 3.4. The link-weighted degree of a node (Abbasi, 2013) In (Abbasi, 2013) the author studied undirected weighted networks and introduced the neighbourhood-weighted degree of node m considered as node n’s neighbour, denoted as wD(n,m), as the product of the degree of node m and the weight of the link between this neighbour and node n. If the network is unweighted, wD(n,m) is the degree of node m if node m is linked to node n; otherwise wD(n,m) is equal to zero. 3.5. Abbasi’s Hw-degree Abbasi (2013) defined the Hw-degree of node n as the largest integer such that the top k neighbours of this node have each a wD-value of at least k. This value k is then called the Hw-degree. 3.6. The link-weighted strength of a node The link-weighted strength of node m considered as node n’s neighbour, denoted as wS(n,m), is the product of the strength of node m and the weight of the link between this neighbour and node n. If the network is unweighted, wS(n,m) is the degree of node m if node m is linked to node n; otherwise wS(n,m) is equal to zero. 3.7. The Hw-strength Yan et al. (2013) defined the Hw-degree of node n as the largest integer such that the top k neighbours of this node have each a wS-value of at least k. This value k is then called the wS-degree of node n. Actually, Yan et al. (2013) defined this notion in a collaboration network and denoted it by c(n). Of course their notion can be applied in any weighted network. In the next section we show how all these notions can be unified using Zipf lists. 4. Zipf lists Consider a set S of elements each characterized by a natural number, referred to as their magnitude. Elements in S are then ranked according to these magnitudes. Elements with the same magnitude are ranked in any order, but with a different rank. In Rousseau, Guns, and Liu (2008) such a list was referred to as a Zipf list and ranks are called Zipf ranks. The h-index of a Zipf list is defined as the highest rank such that the magnitude corresponding to Zipf rank h is at least equal to h. It is clear that one can determine for each Zipf list not only an h-index but also a g-index (Egghe, 2006a, 2006b), an a-index (Jin, 2006) and an R2 -index (Jin, Liang, Rousseau, & Egghe, 2007). It is now clear how the measures mentioned above can be considered as h-indices of appropriate Zipf lists. First we consider nodes in n’s neighbourhood. 4.1. The lobby index One associates to each node in N(n) its degree centrality. Then nodes are ranked according to degree centrality leading to a Zipf list, with degree centralities as magnitudes. The h-index of this list is the lobby index of node n. 4.2. The w-lobby index One associates to each node in N(n) its node strength. Then nodes in N(n) are ranked according to their node strength leading to a Zipf list, with node strength as magnitudes. The h-index of this list is the w-lobby index of node n. 4.3. Abbasi’s Hw-degree We associate to each node m in N(n) the product of its degree and the weight of the link connecting m with n, this is wD(n,m). Using this product as m’s magnitude and calculating the h-index of the corresponding Zipf list leads to Abbasi’s Hw-degree.
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4.4. The hw-strength One associates to each node m in N(n) its magnitude wS(n,m). Again, using wS(n,m) as m’s magnitude, the corresponding h-index of this Zipf list is the Hw-strength of the ego node n. As mentioned above we can not only associate an h-index to each Zipf list but also a g-index, an a-index and an R2 (or an R) index. Some of these have been introduced explicitly in Abbasi (2013), but other ones have never been proposed. As this is straightforward once Zipf lists are introduced, we do not discuss this further. In all these cases we associated a magnitude to a node in n’s neighbourhood. It is, however, equally possible to associate a magnitude to each link adjacent to n, leading to yet another Zipf list. 4.5. The h-degree The h-degree can be expressed using a Zipf list. Indeed links are ranked according to their weight (playing the role of magnitudes). The h-index of this list is the h-degree of node n. 4.6. The lobby index as an h-degree If we make the neighbourhood of node n into a (locally) weighted network by assigning the degree of a neighbouring node to the link connecting this node to node n, then the h-degree of n is equal to its lobby index. Of course, the magnitudes associated to a neighbouring node can equally be assigned to the link between node n and this neighbouring node. This leads to a reweighting of links. Using the new weights as magnitudes for a Zipf list shows that the w-lobby index, Abbasi’s hw-degree and the Hw-strength can also be considered as h-indices of Zipf list associated with links (and not only as h-indices associated with nodes). In this way the notion of an h-degree becomes another unifying concept. 5. Indicators in directed networks Already in (Zhao et al., 2011) the authors pointed out that their definition could easily be adapted to the case of directed weighted networks, leading to an IN h-degree and an OUT h-degree. This observation was elaborated in (Zhao & Ye, 2012). For clarity we formulate the definition of the IN and OUT h-degree in a directed network. Definition. The IN h-degree (dh − ) of node n in a directed weighted network is equal to dIN-h (n), if dIN-h (n) is the largest natural number such that n has dIN-h (n) inlinks each with strength at least equal to dIN-h (n). The set of links contributing to the IN h-degree of node n (consisting of dIN-h (n) links) is called the IN-h-degree core of this node. Replacing the word IN by the word OUT yields the definition of the OUT h-degree and the OUT h-degree core. Besides the lobby index in an undirected, unweighted network one may define an IN- and an OUT-lobby index in a directed network. The IN-lobby index of node n is equal to the largest integer l− (n) such that node n has at least l− (n) in-neighbours with at least indegree centrality l− (n). Similarly we define the OUT-lobby index by replacing IN by OUT. 6. The standard h-index as a network indicator The procedure applied in the previous sections leads to the question: Can the standard h-index (Hirsch, 2005) be described as an h-degree in some network? This is possible indeed. Consider a directed network with an author as its ego (node A). A directed link connects the author to all his/her articles with the direction going from the article to the author. Other links in the network are ‘cites’ links between articles, e.g. a link exists between m (an article, co-authored by A) and article p if p cites m. Hence this is a specific directed network. The IN-degree of such a neighbouring node m is equal to the number of citations received by m. Hence, A’s lobby index in this network is equal to his/her (standard) h-index. Consider now this same network but assign the IN-degree of node p to the link that connects p with m. Then the h-index of this Zipf list of magnitudes leads to the hl -index as introduced in (Zhai, Yan, & Zhu, 2014). Finally, we describe another way to obtain the standard h-index as an h-degree. This approach is meaningful when one wants to study several egos at the same time. Consider a bipartite network consisting of articles and authors. Author A and article a are linked if author A is the author, or one of the authors, of article a. Article a is linked to each of its authors. Links are weighted by the number of received citations (during a given citation window). Clearly, the h-degree of author A, denoted as hD (A), is equal to the standard h-index of A, see Fig. 1 upper row. We note that, in this approach, we have to allow links with zero weight, corresponding to articles with zero citations. The h-degree of articles, shown in the lower row of this weighted bipartite network is equal to the number of authors (if the number of received citations is at least equal to the number of authors), and is equal to the number of received citations (possibly zero) in the other cases.
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Fig. 1. A specific network in which the classical h-index is obtained as an h-degree.
7. Conclusion The concepts of Zipf lists, Zipf ranks and the h-index of a Zipf list can be used to describe most h-type indices via magnitudes of nodes or magnitudes of links. The main theoretical contributions of this article are (1) it provides a unification of different h-type indices introduced previously; (2) it shows that it is possible to shift the focus from h-indices to h-degrees; (3) informetric concepts such as Zipf lists and Zipf ranks can be placed in the framework of network analysis. We suggest further investigations to develop theoretical models based on this framework. Acknowledgements Research by Ronald Rousseau is supported by the Natural Science Foundation of China (NSFC) grants no. 71173154 and 71173185. The authors thank Fred Y. Ye for useful discussions and two anonymous reviewers for correcting errors in an earlier submission. References Abbasi, A. (2013). h-Type hybrid centrality measures for weighted networks. Scientometrics, 96(2), 633–640. Barrat, A., Barthélemy, M., Pastor-Satorras, R., & Vespignani, A. (2004). The architecture of complex weighted networks. Proceedings of the National Academy of Sciences of the United States of America, 101(11), 3747–3752. Bonacich, P. (1987). Power and centrality: A family of measures. American Journal of Sociology, 92(5), 1170–1182. Egghe, L. (2006a). An improvement of the H-index: The G-index. ISSI Newsletter, 2(1), 8–9. Egghe, L. (2006b). Theory and practise of the g-index. Scientometrics, 69(1), 131–152. Glänzel, W. (2012). The role of core documents in bibliometric network analysis and their relation with h-type indices. Scientometrics, 93(1), 113–123. Hirsch, J. E. (2005). An index to quantify an individual’s scientific research output. Proceedings of the National Academy of Sciences of the United States of America, 102(46), 16569–16572. Jin, B. H. (2006). H-index: An evaluation indicator proposed by scientist. Science Focus, 1(1), 8–9 (in Chinese). Jin, B. H., Liang, L. M., Rousseau, R., & Egghe, L. (2007). The R- and AR-indices: Complementing the h-index. Chinese Science Bulletin, 52(6), 855–863. Katz, L. (1953). A new status index derived from sociometric data analysis. Psychometrika, 18(1), 39–43. Korn, A., Schubert, A., & Telcs, A. (2009). Lobby index in networks. Physica A, 388(11), 2221–2226. Otte, E., & Rousseau, R. (2002). Social network analysis: A powerful strategy, also for the information sciences. Journal of Information Science, 28(6), 441–453. Rousseau, R. (2012). Comments on “A Hirsch-type index of co-author partnership ability”. Scientometrics, 91(1), 309–310. Rousseau, R., Guns, R., & Liu, Y. X. (2008). The h-index of a conglomerate. Cybermetrics, 12(1) [paper 2]. http://cybermetrics.cindoc.csic.es/articles/ v12i1p2.html Schubert, A. (2012a). A Hirsch-type index of co-author partnership ability. Scientometrics, 91(1), 303–308. Schubert, A. (2012b). Jazz discometrics – A network approach. Journal of Informetrics, 6(4), 480–484. Schubert, A., & Glänzel, W. (2007). A systematic analysis of Hirsch-type indices for journals. Journal of Informetrics, 1(2), 179–184. Schubert, A., Korn, A., & Telcs, A. (2009). Hirsch-type indices for characterizing networks. Scientometrics, 78(2), 375–382. Wasserman, S., & Faust, K. (1994). Social network analysis: Methods and applications. Cambridge: Cambridge University Press. Yan, X. B., Zhai, L., & Fan, W. G. (2013). C-index: A weighted network node centrality measure for collaboration competence. Journal of Informetrics, 7(1), 223–239. Zhai, L., Yan, X. B., & Zhu, B. (2014). The Hl -index: Improvement of H-index based on quality of citing papers. Scientometrics, 98(2), 1021–1031. Zhao, S. X., Rousseau, R., & Ye, F. Y. (2011). h-Degree as a basic measure in weighted network. Journal of Informetrics, 5(4), 668–677. Zhao, S. X., & Ye, F. Y. (2012). Exploring the directed h-degree in directed weighted networks. Journal of Informetrics, 6(4), 619–630. Zhao, S. X., Zhang, P. L., Li, J., Tan, A. M., & Ye, F. Y. (2014). Abstracting the core subnet of weighted networks based on link strengths. Journal of the Association for Information Science and Technology, 65(5), 984–994.
Contents lists available at ScienceDirect
Journal of Informetrics journal homepage: www.elsevier.com/locate/joi
A general conceptual framework for characterizing the ego in a network Ronald Rousseau a,b , Star X. Zhao c,∗ a b c
Universiteit Antwerpen (UA), IBW, Stadscampus, Venusstraat 35, 2000 Antwerpen, Belgium KU Leuven, Department of Mathematics, 3000 Leuven, Belgium Department of Information Science, Business School, East China Normal University, 200241 Shanghai, China
a r t i c l e
i n f o
Article history: Received 13 September 2014 Received in revised form 7 December 2014 Accepted 8 December 2014 Keywords: Network analysis h-Index h-Degree a-Index g-Index Zipf list
a b s t r a c t In this contribution we consider one particular node in a network, referred to as the ego. We combine Zipf lists and ego measures to put forward a conceptual framework for characterizing this particular node. In this framework we unify different forms of h-indices, in particular the h-degree, introduced in the literature. Similarly, different forms of the gindex, the a-index and the R-index are unified. We focus on the pure mathematical and logical concepts, referring to the existing literature for practical examples. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction An interesting recent development in network analysis is the emergence of h-type network measures such as the lobby index (Korn, Schubert, & Telcs, 2009), the h-degree (Zhao, Rousseau, & Ye, 2011), the partnership ability index (Schubert, 2012a), the Hw-degree (Abbasi, 2013) and the C-index (Yan, Zhai, & Fan, 2013). These indicators provide new approaches for exploring networks. Moreover, empirical investigations (Schubert, 2012a; Zhao, Zhang, Li, Tan, & Ye, 2014) show that the Schubert-Glänzel model (2007) for the h-index is applicable in networks. In this contribution we focus on one particular node in a network, referred to as the ego. As such our approach differs from e.g. (Glänzel, 2012) where the main focus is on the network as a whole. We combine Zipf lists, defined further on, and ego measures to put forward a conceptual framework for characterizing this particular node in networks. In this framework, also the original h-index can be converted into an h-degree. In this way we unify approaches introduced by the authors mentioned above and offer a birds-eye-view on the topic of h-type indicators as applied to networks. Focus is on the pure mathematical and logical concepts, referring the reader for practical examples to the existing literature.
∗ Corresponding author at: Room 437, Building of Law & Business, 500 Dongchuan Rd. (Minhang Campus of East China Normal University), 200241 Shanghai, China. Tel.: +86 15800587558. E-mail addresses: [email protected], [email protected] (R. Rousseau), [email protected] (S.X. Zhao). http://dx.doi.org/10.1016/j.joi.2014.12.002 1751-1577/© 2014 Elsevier Ltd. All rights reserved.
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2. The ego in an undirected, unweighted network In this section we consider an undirected, unweighted network and consider different ways to characterize the ego, node n, as a function of its immediate neighbourhood. Well-known centrality measures such as closeness centrality, eccentricity, betweenness centrality, Katz’ influence and eigenvector centrality are not considered as for their calculation (extremely small networks being an exception) one needs information beyond a node’s immediate neighbourhood (Bonacich, 1987; Katz, 1953; Otte & Rousseau, 2002; Wasserman & Faust, 1994). The first-order neighbourhood of node n is denoted as N1 (n) and consists of all nodes linked to n. Such nodes are referred to as first-order neighbours. The second-order neighbourhood of n consists of all nodes linked to a first order neighbour, node n and all first-order neighbours being excluded. This set is denoted as N2 (n). When it is clear that we are only using first-order neighbours, we denote the first-order neighbourhood of node n as N(n). 2.1. A binary measure The binary measure c(n) takes the value 1 if node n has at least one neighbour (hence one link) and the value 0 if it has not. This binary measure distinguishes between unconnected singletons and nodes that are connected to a least one other node. 2.2. Degree centrality (Wasserman & Faust, 1994) Degree centrality of node n, denoted as deg(n), is equal to the number of nodes in node n’s first-order neighbourhood; hence it is equal to the number of links adjacent to n. 2.3. The lobby index Schubert, Korn, and Telcs (2009) defined the lobby index of node n in an undirected unweighted network as the largest integer l(n) such that node n has at least l(n) neighbours with at least degree centrality l(n). 2.4. The second-order lobby index The second-order lobby index is defined in a similar way as the (first-order) lobby index. The only difference is that one uses N2 (n) instead of N1 (n). Clearly it is possible to define k-th order lobby indices, but as we do not see any application of higher order lobby indices we do not go into this. 3. The ego in an undirected, weighted network In a weighted network a weight is associated with each link. For simplicity we assume that this weight is a natural number, different from zero (for now). 3.1. Node strength The node strength of a node in an undirected weighted network is defined as the sum of the strengths (or weights) of all its links (Barrat, Barthélemy, Pastor-Satorras, & Vespignani, 2004). This is denoted as dS (n). When all weights are equal to 1 (corresponding to the unweighted case) the node strength becomes the degree centrality of the node. 3.2. A node’s h-degree Zhao et al. (2011) introduced the h-degree of a node in a weighted undirected network as follows. Definition. The h-degree (dh ) of node n in a undirected weighted network is equal to dh (n), if dh (n) is the largest natural number such that n has dh (n) links each with strength at least equal to dh (n). If all weights are one or zero (zero, only when there is no link) then the h-degree coincides with the binary measure c(n). In 2012 Schubert proposed the partnership ability index, denoted as (Schubert, 2012a). This notion was introduced in a co-authorship network and defined as the largest natural number P such that an actor has at least P partners with whom he/she had at least P joint articles. It was pointed out that this definition could be adapted to other networks such as movie actor networks, sexual encounter networks, and cooperation between jazz musicians (Schubert, 2012a, 2012b). It is clear that all these cases are just illustrations of the h-degree in an undirected network (Rousseau, 2012; Schubert, 2012a). Indeed, it suffices using the number of joint articles, joint movies, sexual encounters and joint recording sessions and records as weights.
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3.3. The w-lobby index (Zhao et al., 2011) This definition extends the lobby index to the case of a weighted undirected network: the w-lobby index (weighted network lobby index) of node n, denoted as lw (n) is defined as the largest integer k such that node n has at least k neighbours with node strength at least k. Clearly, if links are unweighted, then the w-lobby index coincides with the original lobby index. 3.4. The link-weighted degree of a node (Abbasi, 2013) In (Abbasi, 2013) the author studied undirected weighted networks and introduced the neighbourhood-weighted degree of node m considered as node n’s neighbour, denoted as wD(n,m), as the product of the degree of node m and the weight of the link between this neighbour and node n. If the network is unweighted, wD(n,m) is the degree of node m if node m is linked to node n; otherwise wD(n,m) is equal to zero. 3.5. Abbasi’s Hw-degree Abbasi (2013) defined the Hw-degree of node n as the largest integer such that the top k neighbours of this node have each a wD-value of at least k. This value k is then called the Hw-degree. 3.6. The link-weighted strength of a node The link-weighted strength of node m considered as node n’s neighbour, denoted as wS(n,m), is the product of the strength of node m and the weight of the link between this neighbour and node n. If the network is unweighted, wS(n,m) is the degree of node m if node m is linked to node n; otherwise wS(n,m) is equal to zero. 3.7. The Hw-strength Yan et al. (2013) defined the Hw-degree of node n as the largest integer such that the top k neighbours of this node have each a wS-value of at least k. This value k is then called the wS-degree of node n. Actually, Yan et al. (2013) defined this notion in a collaboration network and denoted it by c(n). Of course their notion can be applied in any weighted network. In the next section we show how all these notions can be unified using Zipf lists. 4. Zipf lists Consider a set S of elements each characterized by a natural number, referred to as their magnitude. Elements in S are then ranked according to these magnitudes. Elements with the same magnitude are ranked in any order, but with a different rank. In Rousseau, Guns, and Liu (2008) such a list was referred to as a Zipf list and ranks are called Zipf ranks. The h-index of a Zipf list is defined as the highest rank such that the magnitude corresponding to Zipf rank h is at least equal to h. It is clear that one can determine for each Zipf list not only an h-index but also a g-index (Egghe, 2006a, 2006b), an a-index (Jin, 2006) and an R2 -index (Jin, Liang, Rousseau, & Egghe, 2007). It is now clear how the measures mentioned above can be considered as h-indices of appropriate Zipf lists. First we consider nodes in n’s neighbourhood. 4.1. The lobby index One associates to each node in N(n) its degree centrality. Then nodes are ranked according to degree centrality leading to a Zipf list, with degree centralities as magnitudes. The h-index of this list is the lobby index of node n. 4.2. The w-lobby index One associates to each node in N(n) its node strength. Then nodes in N(n) are ranked according to their node strength leading to a Zipf list, with node strength as magnitudes. The h-index of this list is the w-lobby index of node n. 4.3. Abbasi’s Hw-degree We associate to each node m in N(n) the product of its degree and the weight of the link connecting m with n, this is wD(n,m). Using this product as m’s magnitude and calculating the h-index of the corresponding Zipf list leads to Abbasi’s Hw-degree.
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4.4. The hw-strength One associates to each node m in N(n) its magnitude wS(n,m). Again, using wS(n,m) as m’s magnitude, the corresponding h-index of this Zipf list is the Hw-strength of the ego node n. As mentioned above we can not only associate an h-index to each Zipf list but also a g-index, an a-index and an R2 (or an R) index. Some of these have been introduced explicitly in Abbasi (2013), but other ones have never been proposed. As this is straightforward once Zipf lists are introduced, we do not discuss this further. In all these cases we associated a magnitude to a node in n’s neighbourhood. It is, however, equally possible to associate a magnitude to each link adjacent to n, leading to yet another Zipf list. 4.5. The h-degree The h-degree can be expressed using a Zipf list. Indeed links are ranked according to their weight (playing the role of magnitudes). The h-index of this list is the h-degree of node n. 4.6. The lobby index as an h-degree If we make the neighbourhood of node n into a (locally) weighted network by assigning the degree of a neighbouring node to the link connecting this node to node n, then the h-degree of n is equal to its lobby index. Of course, the magnitudes associated to a neighbouring node can equally be assigned to the link between node n and this neighbouring node. This leads to a reweighting of links. Using the new weights as magnitudes for a Zipf list shows that the w-lobby index, Abbasi’s hw-degree and the Hw-strength can also be considered as h-indices of Zipf list associated with links (and not only as h-indices associated with nodes). In this way the notion of an h-degree becomes another unifying concept. 5. Indicators in directed networks Already in (Zhao et al., 2011) the authors pointed out that their definition could easily be adapted to the case of directed weighted networks, leading to an IN h-degree and an OUT h-degree. This observation was elaborated in (Zhao & Ye, 2012). For clarity we formulate the definition of the IN and OUT h-degree in a directed network. Definition. The IN h-degree (dh − ) of node n in a directed weighted network is equal to dIN-h (n), if dIN-h (n) is the largest natural number such that n has dIN-h (n) inlinks each with strength at least equal to dIN-h (n). The set of links contributing to the IN h-degree of node n (consisting of dIN-h (n) links) is called the IN-h-degree core of this node. Replacing the word IN by the word OUT yields the definition of the OUT h-degree and the OUT h-degree core. Besides the lobby index in an undirected, unweighted network one may define an IN- and an OUT-lobby index in a directed network. The IN-lobby index of node n is equal to the largest integer l− (n) such that node n has at least l− (n) in-neighbours with at least indegree centrality l− (n). Similarly we define the OUT-lobby index by replacing IN by OUT. 6. The standard h-index as a network indicator The procedure applied in the previous sections leads to the question: Can the standard h-index (Hirsch, 2005) be described as an h-degree in some network? This is possible indeed. Consider a directed network with an author as its ego (node A). A directed link connects the author to all his/her articles with the direction going from the article to the author. Other links in the network are ‘cites’ links between articles, e.g. a link exists between m (an article, co-authored by A) and article p if p cites m. Hence this is a specific directed network. The IN-degree of such a neighbouring node m is equal to the number of citations received by m. Hence, A’s lobby index in this network is equal to his/her (standard) h-index. Consider now this same network but assign the IN-degree of node p to the link that connects p with m. Then the h-index of this Zipf list of magnitudes leads to the hl -index as introduced in (Zhai, Yan, & Zhu, 2014). Finally, we describe another way to obtain the standard h-index as an h-degree. This approach is meaningful when one wants to study several egos at the same time. Consider a bipartite network consisting of articles and authors. Author A and article a are linked if author A is the author, or one of the authors, of article a. Article a is linked to each of its authors. Links are weighted by the number of received citations (during a given citation window). Clearly, the h-degree of author A, denoted as hD (A), is equal to the standard h-index of A, see Fig. 1 upper row. We note that, in this approach, we have to allow links with zero weight, corresponding to articles with zero citations. The h-degree of articles, shown in the lower row of this weighted bipartite network is equal to the number of authors (if the number of received citations is at least equal to the number of authors), and is equal to the number of received citations (possibly zero) in the other cases.
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Fig. 1. A specific network in which the classical h-index is obtained as an h-degree.
7. Conclusion The concepts of Zipf lists, Zipf ranks and the h-index of a Zipf list can be used to describe most h-type indices via magnitudes of nodes or magnitudes of links. The main theoretical contributions of this article are (1) it provides a unification of different h-type indices introduced previously; (2) it shows that it is possible to shift the focus from h-indices to h-degrees; (3) informetric concepts such as Zipf lists and Zipf ranks can be placed in the framework of network analysis. We suggest further investigations to develop theoretical models based on this framework. Acknowledgements Research by Ronald Rousseau is supported by the Natural Science Foundation of China (NSFC) grants no. 71173154 and 71173185. The authors thank Fred Y. Ye for useful discussions and two anonymous reviewers for correcting errors in an earlier submission. References Abbasi, A. (2013). h-Type hybrid centrality measures for weighted networks. Scientometrics, 96(2), 633–640. Barrat, A., Barthélemy, M., Pastor-Satorras, R., & Vespignani, A. (2004). The architecture of complex weighted networks. Proceedings of the National Academy of Sciences of the United States of America, 101(11), 3747–3752. Bonacich, P. (1987). Power and centrality: A family of measures. American Journal of Sociology, 92(5), 1170–1182. Egghe, L. (2006a). An improvement of the H-index: The G-index. ISSI Newsletter, 2(1), 8–9. Egghe, L. (2006b). Theory and practise of the g-index. Scientometrics, 69(1), 131–152. Glänzel, W. (2012). The role of core documents in bibliometric network analysis and their relation with h-type indices. Scientometrics, 93(1), 113–123. Hirsch, J. E. (2005). An index to quantify an individual’s scientific research output. Proceedings of the National Academy of Sciences of the United States of America, 102(46), 16569–16572. Jin, B. H. (2006). H-index: An evaluation indicator proposed by scientist. Science Focus, 1(1), 8–9 (in Chinese). Jin, B. H., Liang, L. M., Rousseau, R., & Egghe, L. (2007). The R- and AR-indices: Complementing the h-index. Chinese Science Bulletin, 52(6), 855–863. Katz, L. (1953). A new status index derived from sociometric data analysis. Psychometrika, 18(1), 39–43. Korn, A., Schubert, A., & Telcs, A. (2009). Lobby index in networks. Physica A, 388(11), 2221–2226. Otte, E., & Rousseau, R. (2002). Social network analysis: A powerful strategy, also for the information sciences. Journal of Information Science, 28(6), 441–453. Rousseau, R. (2012). Comments on “A Hirsch-type index of co-author partnership ability”. Scientometrics, 91(1), 309–310. Rousseau, R., Guns, R., & Liu, Y. X. (2008). The h-index of a conglomerate. Cybermetrics, 12(1) [paper 2]. http://cybermetrics.cindoc.csic.es/articles/ v12i1p2.html Schubert, A. (2012a). A Hirsch-type index of co-author partnership ability. Scientometrics, 91(1), 303–308. Schubert, A. (2012b). Jazz discometrics – A network approach. Journal of Informetrics, 6(4), 480–484. Schubert, A., & Glänzel, W. (2007). A systematic analysis of Hirsch-type indices for journals. Journal of Informetrics, 1(2), 179–184. Schubert, A., Korn, A., & Telcs, A. (2009). Hirsch-type indices for characterizing networks. Scientometrics, 78(2), 375–382. Wasserman, S., & Faust, K. (1994). Social network analysis: Methods and applications. Cambridge: Cambridge University Press. Yan, X. B., Zhai, L., & Fan, W. G. (2013). C-index: A weighted network node centrality measure for collaboration competence. Journal of Informetrics, 7(1), 223–239. Zhai, L., Yan, X. B., & Zhu, B. (2014). The Hl -index: Improvement of H-index based on quality of citing papers. Scientometrics, 98(2), 1021–1031. Zhao, S. X., Rousseau, R., & Ye, F. Y. (2011). h-Degree as a basic measure in weighted network. Journal of Informetrics, 5(4), 668–677. Zhao, S. X., & Ye, F. Y. (2012). Exploring the directed h-degree in directed weighted networks. Journal of Informetrics, 6(4), 619–630. Zhao, S. X., Zhang, P. L., Li, J., Tan, A. M., & Ye, F. Y. (2014). Abstracting the core subnet of weighted networks based on link strengths. Journal of the Association for Information Science and Technology, 65(5), 984–994.