Posetic Tools in the Social Sciences: A Tutorial Exposition
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3.1.2 Dominance Eigenvector
This scoring procedure is based on the Singular Value Decomposition (Meyer 2000)
of the mutual ranking probability matrix M, associated to the input poset π . M
is a non-negative matrix and so, by the Perron-Frobenius (Meyer 2000), its first
right singular vector v = (v 1 , . . . , v n ), i.e. the eigenvector 1 of M T M relative to the
greatest eigenvalue, has strictly positive components. In addition, it can be proven
that such components are such that x i x j in π implies v i < v j . Therefore, the
score function s(·)
s : π → R
+
: x i → v i
is strictly order preserving and can be used to rank the elements of the input poset.
Vector v is a linear combination of the rows of M, so the score associated to element
x is a weighted average of the probabilities that x dominates poset elements. By the
properties of the Singular Value Decomposition, vector v has the optimal property
to provide the best uni-dimensional approximation to the mutual ranking probability
matrix; more precisely, it turns out that the rows of the rank-one matrix ˆ
M which
best approximates matrix M in the Euclidean norm (here called, Frobenius norm)
are proportional to v.
Both the average rank and the dominance eigenvector preserve linear orders,
i.e. if the input poset is a linear order λ, then the final ranking is λ itself. This
natural property is not shared by other scoring functions proposed in the literature
(Todeschini et al. 2015; Saaty and Hu 1998), which extract eigenvectors directly
from the mutual ranking probability M and not from matrix M T M. Notice also that,
in general, scoring functions can produce ties, whenever different profiles occupy
“equivalent” positions in the input poset.
3.1.3 Real Example
Table 1 reports the values of three economic indicators for EU-28 member states
(year 2017). These indicators are numerical, but refer to different features of national
economies; combining them into a composite index can be misleading and so we
keep them separate and represent the dataset as a poset, whose Hasse diagram is
drawn in Fig. 1. Quite naturally, country j dominates country i in the diagram, if
1 Let A be a square matrix; a vector x is called eigenvector of A relative to the eigenvalue a if it
holds Ax = ax, where a is a real number. Eigenvectors, when they exist, provide deep information
on the structure of the input matrix and are often used in multivariate statistics, to produce optimal
data synthesis.
225
3.1.2 Dominance Eigenvector
This scoring procedure is based on the Singular Value Decomposition (Meyer 2000)
of the mutual ranking probability matrix M, associated to the input poset π . M
is a non-negative matrix and so, by the Perron-Frobenius (Meyer 2000), its first
right singular vector v = (v 1 , . . . , v n ), i.e. the eigenvector 1 of M T M relative to the
greatest eigenvalue, has strictly positive components. In addition, it can be proven
that such components are such that x i x j in π implies v i < v j . Therefore, the
score function s(·)
s : π → R
+
: x i → v i
is strictly order preserving and can be used to rank the elements of the input poset.
Vector v is a linear combination of the rows of M, so the score associated to element
x is a weighted average of the probabilities that x dominates poset elements. By the
properties of the Singular Value Decomposition, vector v has the optimal property
to provide the best uni-dimensional approximation to the mutual ranking probability
matrix; more precisely, it turns out that the rows of the rank-one matrix ˆ
M which
best approximates matrix M in the Euclidean norm (here called, Frobenius norm)
are proportional to v.
Both the average rank and the dominance eigenvector preserve linear orders,
i.e. if the input poset is a linear order λ, then the final ranking is λ itself. This
natural property is not shared by other scoring functions proposed in the literature
(Todeschini et al. 2015; Saaty and Hu 1998), which extract eigenvectors directly
from the mutual ranking probability M and not from matrix M T M. Notice also that,
in general, scoring functions can produce ties, whenever different profiles occupy
“equivalent” positions in the input poset.
3.1.3 Real Example
Table 1 reports the values of three economic indicators for EU-28 member states
(year 2017). These indicators are numerical, but refer to different features of national
economies; combining them into a composite index can be misleading and so we
keep them separate and represent the dataset as a poset, whose Hasse diagram is
drawn in Fig. 1. Quite naturally, country j dominates country i in the diagram, if
1 Let A be a square matrix; a vector x is called eigenvector of A relative to the eigenvalue a if it
holds Ax = ax, where a is a real number. Eigenvectors, when they exist, provide deep information
on the structure of the input matrix and are often used in multivariate statistics, to produce optimal
data synthesis.