## 3.3 The Integrated Centrality Measure

The classical graph center [4] is the vertex(es) with the lowest eccentricity *e*(*i*), that is, the lowest maximum vertex distance:

This definition often produces multiple central points and can also be considered as the first step in a hierarchical definition for a graph center [33, 34], the next hierarchical criterion (applied at the same minimal eccentricity) being the minimum vertex distance:

The vertex distance itself defines a different graph center called the *median* [35]. Its inverse has been used to define a centrality measure in the social sciences called the *closeness centrality*, *CC* [36]:

where *V* is the number of graph vertices.

The *betweenness centrality* [37, 38] of a vertex *i*, *BC*(*i*), is defined as the fraction of the shortest paths that traverse that vertex:

The idea behind this centrality concept is that vertices that occur more frequently on the shortest paths between other vertices have higher centrality than those that do not.

The simplest centrality measure is based on the vertex degree and is called the *degree centrality* [37]. The ...

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