A general conceptual framework for characterizing the ego in a network
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 et al., 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.
Section snippets
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
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).
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
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
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
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
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.
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