Structural Analysis of Networks - Methods for Multilevel Analysis and Visualisation of Geographical Networks - page 69

Fig. 5.2 The figure shows how values of the ( left ) stress centralities Eq. ( 5.7 )and( right )
betweenness Eq. ( 5.8 ) are distributed over the nodes of a typical small world graph. Note how
both metrics compare well. The node color changes from light orange to purple as the values
increase. The node size also correlates with the node values (Color figure online)

all distance-based metrics described in the preceding paragraph. In fact, distances
between nodes do not fully describe how nodes are connected together and do not
explicitly indicate whether actors have an essential role in distributing information
through the network. Roughly speaking, focusing on the notion of paths , we aim to
identify nodes that play an essential role in making information available to others,
as opposed to the preceding paragraph, in which we tried to identify nodes that had
easy access to information. That being said, distance-based and/or shortest path-
based metrics can nevertheless be correlated, as we shall describe.
More precisely, let

σ vw denote the number of shortest paths connecting two nodes

v and w .Weset 1 by convention. This number may be greater than '1':

imagine a cycle of size 2 n for which this number is equal to 2 when nodes sit at
distance n (that is, when they are one diameter away from each other). Let us also
write

σ vv =

for the number of shortest paths connecting v and w and going through
u (this number may be equal to 0). Shimbel ( 1953 ) defined the stress centrality of a
node as:

σ vw ( u )

)= ∑
v

S G ( u

V σ vw (

u ) . (5.7)

, w ∈

Anthonisse ( 1971 ) and Freeman ( 1979 ) independently considered a ratio involv-
ing both σ vw ( u ) and σ vw , which is now known as a node's betweenness centrality :

σ ( u ) vw

)= ∑
v , w ∈ V

B G ( u . (5.8)

σ vw

Figure 5.2 shows both metrics mapped in terms of color and node size. As can be
observed, these two variations correlate well, which can be confirmed, for instance,
by examining the histogram of values.