Geoscience Reference
In-Depth Information
Supermatrix of a Network
The typical entry of this supermatrix is
Entries in Supermatrix of a Network
2
3
ðÞ
i1
W
jn
j
W
j
ðÞ
i1
W
j
ðÞ
4
5
i1
ðÞ
i1
W
jn
j
W
j
ðÞ
i2
W
j
ðÞ
i2
W
ij
¼
ðÞ
in
i
W
jn
j
W
j
ðÞ
in
i
W
j
ðÞ
in
i
The entries of sub-matrixes in
W
ij
are the ratio scales derived from paired
comparisons performed on the elements within the clusters themselves according
to their influence on each element in another cluster (outer dependence) or elements
in their own cluster (inner dependence). Judgments are elicited from which ratio
scales are derived. The resulting unweighted supermatrix is then transformed into a
matrix where each column sums to unity to generate a stochastic supermatrix. The
derived weights are used to weight the elements of the corresponding column
blocks (cluster) of the supermatrix, resulting in a weighted supermatrix which is
also stochastic. The final ranking is derived from the limiting supermatrix, obtained
by raising the stochastic supermatrix to large powers to read off final priorities, in
which all matrix columns are identical. Each gives the relative priorities of the
