Elsevier

Social Networks

Volume 28, Issue 4, October 2006, Pages 466-484
Social Networks

A Graph-theoretic perspective on centrality

https://doi.org/10.1016/j.socnet.2005.11.005Get rights and content

Abstract

The concept of centrality is often invoked in social network analysis, and diverse indices have been proposed to measure it. This paper develops a unified framework for the measurement of centrality. All measures of centrality assess a node's involvement in the walk structure of a network. Measures vary along four key dimensions: type of nodal involvement assessed, type of walk considered, property of walk assessed, and choice of summary measure. If we cross-classify measures by type of nodal involvement (radial versus medial) and property of walk assessed (volume versus length), we obtain a four-fold polychotomization with one cell empty which mirrors Freeman's 1979 categorization. At a more substantive level, measures of centrality summarize a node's involvement in or contribution to the cohesiveness of the network. Radial measures in particular are reductions of pair-wise proximities/cohesion to attributes of nodes or actors. The usefulness and interpretability of radial measures depend on the fit of the cohesion matrix to the one-dimensional model. In network terms, a network that is fit by a one-dimensional model has a core-periphery structure in which all nodes revolve more or less closely around a single core. This in turn implies that the network does not contain distinct cohesive subgroups. Thus, centrality is shown to be intimately connected with the cohesive subgroup structure of a network.

Introduction

Centrality is a fundamental concept in network analysis. Bavelas, 1948, Bavelas, 1950 and Leavitt (1951) used centrality to explain differential performance of communication networks and network members on a host of variables including time to problem solution, number of errors, perception of leadership, efficiency, and job satisfaction. Their work led to a great deal of experimental, empirical, and theoretical research on the implications of network structure for substantive outcomes, particularly in the context of organizations. Centrality has been used to investigate influence in interorganizational networks (Laumann and Pappi, 1976, Marsden and Laumann, 1977, Galaskiewicz, 1979), power (Burt, 1982, Knoke and Burt, 1983), advantage in exchange networks (Cook et al., 1983, Marsden, 1982), competence in formal organizations (Blau, 1963), employment opportunities (Granovetter, 1974), adoption of innovation (Coleman et al., 1966), corporate interlocks (Mariolis, 1975, Mintz and Schwartz, 1985, Mizruchi, 1982), status in monkey grooming networks (Sade, 1972, Sade, 1989), power in organizations (Brass, 1984) and differential growth rates among medieval cities (Pitts, 1979). In addition, many other studies use well-known measures of centrality but do not identify them as such. For example, researchers working with ego-networks use the term “network size” (Campbell et al., 1986, Deng and Bonacich, 1991) to refer to a variable that in another context we would recognize as degree centrality.

While many measures of centrality have been proposed, the category itself is not well defined beyond general descriptors such as node prominence or structural importance. In addition, people propose all kinds of interpretations of centrality measures, such as (potential for) autonomy, control, risk, exposure, influence, belongingness, brokerage, independence, power and so on. The one thing that all agree on is that centrality is a node-level construct. But what specifically defines the category? What do all centrality measures have in common? Are there any structural properties of nodes that are not measures of centrality?

Sabidussi (1966) tried to provide a mathematical answer to these questions. He suggested a set of criteria that measures must meet in order to qualify as centrality measures. For example, he felt that adding a tie to a node should always increase the centrality of the node, and that adding a tie anywhere in the network should never decrease the centrality of any node. These requirements are attractive: it is easy to see the value of separating measures that are “well-behaved” from measures that behave less intuitively. However, there are problems with Sabidussi's approach. For one thing, it turns out that his criteria eliminate most known measures of centrality, including betweenness centrality. This is clearly unsatisfactory. Furthermore, while his criteria provide some desirable, prescriptive characteristics for a centrality measure, they do not actually attempt to explain what centrality is.

Freeman (1979) provided another approach to answering the ‘what is centrality’ question. He reviewed a number of published measures and reduced them to three basic concepts for which he provided canonical formulations. These were degree, closeness and betweenness. He noted that all three attain their maximum values for the center of a star-shaped network. It can be argued that this property serves as a defining characteristic of proper centrality measures.

Borgatti (2005) has recently proposed a dynamic model-based view of centrality that focuses on the outcomes for nodes in a network where something is flowing from node to node across the edges. He argues that the fundamental questions one wants to ask about individual nodes in the dynamic flow context are (a) how often does traffic flow through a node and (b) how long do things take to get to a node. Once these questions are set, it becomes easier to construct graph-theoretic measures based on the structure of the network that predict the answers to these questions. Hence, in this approach, measures of centrality are cast as predictive models of specific properties of network flows.

In this paper, we present an alternative perspective that eschews the dynamic element and is fundamentally structural in character. It is a graph-theoretic review of centrality measures that classifies measures according to the features of their calculation. Whereas the model-based view is centered on the outcomes of centrality, the graph-theoretic view is centered on the way centrality measures are calculated. In short, the present perspective is a means-based classification rather than the ends-based classification presented by Borgatti (2005).

Section snippets

Terminology

For simplicity (and in accordance with centrality convention), we will assume that all networks on which we might compute centrality measures consist of undirected graphs G(V, E), in which V is a set of nodes (also called vertices, points or actors) and E is a set of edges (also called ties or lines) that connect them. Many centrality measures can be discussed in terms of directed graphs as well, but this topic is not treated here. It will be helpful to represent a graph in terms of its

Comparison of methods

To explain the graph-theoretic perspective, we begin by considering a sample of centrality measures and examining how they are computed. In a process similar to the anthropological technique of componential analysis, we extract dimensions along which measures vary. These are then used to develop a three-way typology of measures. We organize the discussion around the three best-known measures of centrality: degree, closeness and betweenness (Freeman, 1979).

A Typology of measures

It is apparent in this review of measures that all of the measures evaluate a node's involvement in the walk structure of a network. That is, they evaluate the volume or length of walks of some kind that originate, terminate, or pass through a node. Furthermore, all are based on the marginals of an appropriately constructed node-by-node matrix, although the method of calculating marginals can vary from simple sums to averages and weighted averages to harmonic means, and so on. Thus four basic

Radial measures and the core periphery assumption

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

Discussion

The differences between radial and medial measures discussed in the last section suggest that this distinction is more important than the volume versus length distinction. In choosing between volume and length measures, one is choosing between different conceptions of cohesion. It seems plausible to suggest that, for a given theoretical application, it is possible to say that one is better than the other. For example, if one is studying risk of receiving in a timely manner something flowing

Conclusion

Following Sabidussi (1966), we have described the notion of centrality in purely graph-theoretic terms: what all measures of centrality do is assess a node's involvement in the walk structure of a network. This is the graph-theoretic answer to the question ‘What do centrality measures measure?’ We have suggested that centrality measures differ along four key dimensions: choice of summary measure, type of walk considered, property of walk assessed, and type of involvement. The choice of summary

Acknowledgements

We thank Phil Bonacich, Lin Freeman, Noah Friedkin, and Bill Stevenson for extensive comments on this and earlier versions.

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    Earlier versions of this paper were presented at the 1991 NSF Conference on Measurement Theory and Networks (Irvine, CA), 1992 annual meeting of the American Anthropological Association (San Francisco), and the 1993 Sunbelt XII International Social Network Conference (Tampa, FL).

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