Page 327 - Becoming Metric Wise

320 Becoming Metric-Wise

Rousseau, 2002). Although all these measures are applied to identify and
characterize key nodes in a network, various centrality measures focus on
different roles played by nodes in a network. The most important central-
ity measures are degree centrality, closeness centrality, and betweenness
centrality. They can all be interpreted as measuring aspects of leadership.
Degree centrality refers to activity, closeness centrality to efficiency, while
betweenness refers to control over the flow in the network.
Degree centrality of a node is defined as the number of neighbors
this node has (in graph-theoretical terminology: the number of edges
adjacent to this node). It is the most basic indicator of a node in a net-
work. In mathematical terms degree centrality, d(i), of node i is defined
as:

X
degðiÞ 5 m ij (10.3)
j
where m ij 5 1 if there is a link between nodes i and j, and m ij 5 0 if there
is no such link. In this context m ii 5 0 for all i. In a coauthor graph the
degree centrality of an author is just the number of authors in the graph
with whom they have coauthored at least one article. The degree central-
ity in an N-node network can be standardized (Freeman, 1979) by divid-
ing by N 2 1: deg S (i) 5 d(i)/(N 2 1).
The distribution of degree centralities (or, in short, degrees) of the
nodes in a network is often examined when studying networks; for many
real-world networks this degree distribution is highly skewed.
Many other parameters such as those based on shortest paths and fun-
damental properties of networks such as scale-free phenomena (Baraba ´si
& Albert, 1999; Baraba ´si et al., 2000) are based on the notion of a degree.
In a directed network one makes a distinction between the in-degree and
the out-degree of a network.
The following measures make use of the distance function.
Closeness centrality of a node (Sabidussi, 1966) is equal to the total
distance (in the graph) of this node from all other nodes. As a mathemati-
cal formula closeness centrality, c(i), of node i can be written as:
X
cðiÞ 5 d G ði; tÞ (10.4)
tAV
where d G (i,t) is the distance from node i to node t. In this form closeness
is an inverse measure of centrality in that a larger value indicates a less
central actor while a smaller value indicates a more central actor. For this