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Operating a similar calculus for the node 5, the result is =
0+0+0+1+0+0+1+0+0+0+1+0+0+1+0,5+0+1
+0+0+1+0,5+1+0+0+1+0,5+1+1+0+0+0+0
= 11,5/32
= 0,334
significantly lower than the previous one as expected.
-
Out Eigenvector centrality
: It is based on the assumption that the value of a single
vertex depends on the values of the neighboring vertices and not only on the position
of a vertex within the graph. The moral is that the popularity of a node depends also
on its proximity to other nodes highly connected.
Mathematica
shows the following
out-eigenvector centrality measure for the graph of our example:
Table 17.4
Out-Eigenvector centrality
Node
Score
1
0.999991
2
0.004339
10
−
8
3
8.16933
×
3.27441
×
10
−
15
4
5
0.0000188273
1.77148
×
10
−
10
6
1,77326
×
10
−
10
7
7.68872
×
10
−
13
8
7.69543
×
10
−
13
9
In the aforementioned example we can see that node 1 has a higher degree than
node 4, but node four has a higher score in closeness and betweenness that node 1.
As suggested by Obietat et al. in [7], we think that it is convenient to combine the
degree, betweenness and closeness centrality measures in order to reach a consensus
centrality measure. Combination is needed because the degree measure only takes
into account the direct connections, ignoring the importance of indirect links. If we
adopt as the consensus measure the arithmetical mean, we can see - attending our
example - that node 1 has a consensus measure: 0
,
+
,
375 (degree)
0
348 (closeness)
+
,
/
=
,
,
+
,
+
,
/
=
,
480.
Thus, node 4 is the node with a highest centrality measure in the graph. Therefore,
the content of node 4 complements the prior cause in the why-question explana-
tion. This would be enough to answer to a lay audience. But if people involved
in the inquiry are specialized, other than centrality nodes should provide technical
information. Eigenvector centrality permits to select nodes not only because they
are central, but because they are highly connected to central nodes. In our example,
nodes with high eigenvalue score are: 1
0
25 (betweenness)
3
0
324 and node 4: 0
25
0
533
0
656
3
0
6. Perhaps we need to per-
form a selection in this set. Node 1 is out as it is the prior cause. Nodes 2 and 3
are before node 4, the central node par excellence. So nodes 2 and 3 are explanans
>
2
>
5
>
3
>