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Fig. 5.1
The figure shows how values of the (
left
) Beauchamp Eq. (
5.2
)and(
right
) integration
Eq. (
5.6
) distribute over nodes of a typical small world graph. Note how the Beauchamp
reachability acts as a global measure, whereas the integration metric reveals local features. 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)
Several authors have proposed variations to attempt to measure reachability; all
of these variations use the distances between nodes as a basic component. Valente
and Foreman (
1998
), for instance, define the
integration metric
as:
1
d
G
(
∑
v
∈
V
u
,
v
)
I
G
(
u
)=
.
(5.6)
|
|−
V
1
Just as with Beauchamp's reachability metric, the integration metric lies in the
bounded interval
. At first glance, one may think that these metrics actually
compare or even correlate with each other. Figure
5.1
shows that this is not the
case and illustrates how Beauchamp's closeness centrality is a global measure,
unlike integration, which acts as a local indicator of the degree to which a node
“integrates” in its close-by neighborhood. It is true, however, that the closeness
centrality correlates well with eccentricity.
Dangalchev (
2006
) recently proposed exponentially altering distances before
computing the same sum
[
0
,
1
]
∑
v
∈
V
2
−
d
G
(
u
,
v
)
. Surprisingly, Dangalchev's residual close-
ness strongly correlates with Beauchamps's closeness centrality; this correlation
indicates that the geometric transform has a “globalization effect” on the distances.
Note also that Dangalchev's centrality can be computed even when the graph is
disconnected as 2
−
d
G
(
u
,
v
)
=
0if
d
G
(
u
,
v
)=
∞
.
5.2.2
Shortest Paths
Other authors designed measures that capture the degree to which a node is central
in connecting the other elements of a network. This is somewhat different from
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