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Fig. 12.3
Betweenness centrality index around Bordeaux (
a
) and Lille (
b
), year 1975
to quantify the centrality of a vertex or an edge in a graph. The vertices that
appear on many shortest paths have higher betweenness than those that do not. The
betweenness centrality (
Brandes
,
2001
) of a vertex is as follows:
σ
st
(
v
)
∑
s
=
v
=
t
∈
V
C
B
(
v
)=
σ
st
The value of
σ
st
is the number of shortest paths in the graph between
s
and
t
,and
is the number of shortest paths between
s
and
t
that pass through
the vertex
v
. If the value of the betweenness centrality is high, the node is a
bridge between components in the graph. Otherwise, the node is part of a densely
connected cluster. In the study of the daily network of commuters, the nodes with
high betweenness centrality are municipalities that directly receive workers from
many other municipalities. Consequently, these nodes are points of convergence for
commuters because they are employment centers and major cities in France.
We give edges a color that is based on the following metric: the components
with high betweenness centrality are darker colored while the components with low
betweenness centrality are lighter colored. The examples of Bordeaux and Lille are
developed next (Figs.
12.3
,
12.4
,
12.5
,and
12.6
).
σ
(
v
)
st
12.3.1.2
Strength value
We calculate the strength value (
Auber et al.
,
2003
) for each node and edge in the
graph. Their average of the strength values enable us to know the cohesion in some
sub regions, which in this case are the urban areas. In the following figures, the
strength index has been applied in the major French urban areas such as Marseille
and Lyon, providing us with a visualization of the high cohesion zones that are a
sign of subcenters (Figs.
12.7
,
12.8
,
12.9
,and
12.10
).
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