Introduction

A large number of different centrality measures have been defined in the fields of social science, physics, computer sciences, and so on. By exploiting the structure of a graph, these quantities assign a score to each node of the graph G to reflect the extent to which this node is “central” with respect to G or a subgraph of G, that is, with respect to the communication flowbetween nodes construed in a broad sense. Centrality measures tend to answer the following questions: What is the most representative, or central, node within a given community? How critical is a given node with respect to information flow in a network? Which node is the most peripheral in a social network? Centrality scores attempt to tackle these problems by modeling and quantifying these different, vague, properties of nodes.

In general, these centrality measures are computed on undirected graphs or, when dealing with a directed graph, by ignoring the direction of the edges. They are therefore called “undirectional” [804]. Measures defined on directed graphs – and which are therefore directional – are often called importance or prestige measures, and are discussed in the next chapter. They capture the extent to which a node is “important,” “prominent,” or “prestigious” with respect to the entire directed graph by considering the directed edges as representing some kind of endorsement. Therefore, in this chapter, unless otherwise stated, all networks are considered to be undirected.

As discussed in [469], several attempts have been made to define a typology of centrality measures according to various criteria – for instance a node's involvement in the walk structure of a network; see, for example, [111, 123, 294] for details. In this chapter, only some of the most popular measures are described. For a more detailed account, see, for example, [105, 111, 804].

More precisely, three types of centrality measures are discussed in this chapter:

1. ▸ closeness centrality, quantifying the extent to which a node, or a group of nodes, is central to a given network, that is, its proximity to other nodes in the graph