Page 329 - Becoming Metric Wise

322 Becoming Metric-Wise

Fig. 10.1 presents three simple examples of undirected networks and
differences in their characteristics. All networks have five nodes. We use
adjacency matrices (Table 10.2) to describe the networks. Density and
centrality measures are given in Table 10.3
The density index, D, of this example indicates that the networks (a), (b),
and (c) (considered in this order) become increasingly dense. All centrality
measures show that node u is the center, and that the other nodes become
increasingly central (this is have a larger centrality value) when considering
graphs (a), (b), and (c) (in this order). The relative values also illustrate the
difference between the three centrality measures considered here.

Eigenvector Centrality
Already in 1972, Bonacich (1972) proposed the notion of eigenvector
centrality. Especially during the latest two decades this measure has gained
in popularity, mainly because of its close relation with Google’s
PageRank. Eigenvector centrality values of nodes are obtained as the
components of the principal eigenvector, v, of the adjacency matrix, A,of
the network. Mathematically, eigenvector values are found as the compo-
nents of the vector v, the solution of the vector equation: Av 5 λ v, where
the matrix A is the adjacency matrix of the network, λ (the Greek letter
lambda) is the largest eigenvalue of the network and v is the corresponding
eigenvector. If v is an eigenvector corresponding to the eigenvalue λ,
then cv, with c a positive number is also an eigenvector. Hence one has to
normalize the eigenvector, usually in such a way that the sum of the com-
ponents is one. If a node has a high eigenvector centrality value this usu-
ally points to the fact that it is situated close to other nodes with high
eigenvector centrality values. The eigenvector centrality value can be
interpreted as a measure for the influence of a node. If, for example,
node w influences only one other node, but this node has a very high
influence, then also the influence of node w is high.

The Gefura Measure
There exist centrality measures that take preexisting groupings in a net-
work into account. One of these is the so-called gefura or Q-measure
(Flom et al., 2004; Guns & Rousseau, 2015). The value of this measure
reflects the importance of a node as a bridge between different groups.

Definition: gefura measure for two groups in an undirected network
should better be placed below the Tables 10.2 and 10.3, preceding the
part in which the definition is actually given.