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Network Analysis Literacy pp 243–276Cite as

Centrality Indices

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Part of the book series: Lecture Notes in Social Networks ((LNSN))

Abstract

Centrality indices quantify the importance of a node in a given network, which is often identified with the importance of the corresponding entity in the complex system modeled by the network. As the perceived importance is dependent on the kind of network to be analyzed, different centrality indices have been proposed over the years. This chapter gives a short overview of the most important centrality indices, a characterization of centrality indices based on graph-theory, visualization of centrality indices, and some general schemes of how centrality indices are used to analyze networks with selected examples. The main insight of the chapter is that it is necessary to carefully match a centrality index with the research question of interest to enable a meaningful interpretation of the index’ results.

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Notes

  1. 1.

    One of these is Sabidussi’s definition which requires, among others, that adding an edge to the most central vertex of a graph should result in a graph in which this node is still the most central [44, p. 13–15]. This does not hold for many classic centrality indices, e.g., closeness or eccentricity described below.

  2. 2.

    This index was introduced by Sade as n-path centrality but since n is normally defined as the number of nodes in a graph, Borgatti and Everett called it the k-path centrality; k denotes the maximal length of a path to be considered [45].

  3. 3.

    As \(\alpha \) approaches \(1/\lambda _1\) from below, values of Katz’ centrality will correlate with the eigenvector centrality in most cases [7, p. 556–557], [10, p. 6].

  4. 4.

    The index was introduced by Shimbel as accessibility [46, Eq. 3].

  5. 5.

    Technically, it was first defined by Anthonisse in a technical report as the so-called rush in a network [2]. It was, however, only a technical description of it, without a deduction from or application to a real-world problem.

  6. 6.

    Note that Freeman explicitly excluded the start and target nodes of a shortest path, i.e., \(\sigma _{st}(s):=\sigma _{st}(t)=0\).

  7. 7.

    Note that technically it is a bit more complicated since random walks which walk back and forth over the same node will not contribute to the centrality of this node [41].

  8. 8.

    Borgatti and Everett summarize radial volume or length measures very succinctly in the following way:

    It is apparent that all radial measures are constructed the same way. First one defines an actor-by-actor matrix W that records the number or length of walks of some kind linking every pair of actors. Then one summarizes each row of W by taking some kind of mean or total. Thus, centrality provides an overall summary of a node’s participation in the walk structure of the network. It is a measure of how much of the walk structure is due to a given node. It is quite literally the node’s share of the total volume or length of walks in the network. Thus, the essence of a radial centrality measure is this: radial centrality summarizes a node’s connectedness with the rest of the network [10, p. 12].

    .

  9. 9.

    Other models let the package wait until the best neighbor is free again, which can lead to dead-locks in which two or more nodes wait for each other to be freed. Yet another model lets the packages wander purely at random.

  10. 10.

    Originally shown by Jordan in 1869 [32], re-stated by Hage and Harary [28].

  11. 11.

    The upper diagonal of a matrix comprises all entries A[i, j] where the column counter i is larger than the row counter j.

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  1. TU Kaiserslautern, FB Computer Science, Graph Theory and Analysis of Complex Networks, Gottlieb-Daimler-Str.48, 67331, Kaiserslautern, Germany

    Katharina A. Zweig

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Zweig, K.A. (2016). Centrality Indices. In: Network Analysis Literacy. Lecture Notes in Social Networks. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0741-6_9

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