Page 178 - Becoming Metric Wise

170 Becoming Metric-Wise

M i,j 5 C i,j /S i .If W i is the influence weight of journal i (the new type of
impact factor) then W i is determined by the requirement:
N
X
W i 5 M ji W j (6.11)
j51
hence taking the weights of the journals into account. This may look like a
circular reasoning as the W’s are calculated from the W’s. Yet, this is not
really true. It is known (Langville & Meyer, 2006) that this equality means
that W i is the i-th component of the solution of the matrix equation

t
t
W 5 M W or M W 5 1W (6.12)
The symbol superscript t in this equation represents matrix transposi-
tion, i.e., replacing rows by columns and vice versa. This equation only
determines the direction of the eigenvector W. A unique solution is
determined by an extra—normalization—requirement. Pinski and Narin
normalized in such a way that the average value of the components of W
is equal to 1. An equation of the type (6.12) is called an eigenvector
t
equation. The reason for this name is that matrix M maps the vector W
to a multiple of itself (eigen is German for self ). In this particular case this
multiple, called the eigenvalue, is equal to 1.
Basic mathematical (algebraic) methods cannot solve eigenvector
equations for large matrices. For this reason one applies approximate, iter-
ative methods such as the power method. In its basic approach one pro-
poses a solution for W consisting completely of ones. Substituting these
in the left-hand side of (6.12) leads to a better approximation. Then this
new approximation is substituted in the left-hand side of (6.12) and so
on, until subsequent approximations do not change anymore (more pre-
cisely the first decimals do not change anymore).
In order to apply the Pinski-Narin algorithm to a real network some
adaptations are needed and choices must be made, leading to several var-
iants of the basic algorithm. These include the Eigenfactor Score, the arti-
cle influence score (AIS) and the SCImago Journal Rank. The
Eigenfactor Score is a practical realization (Bergstrom, 2007; Bergstrom
et al., 2008) based on the journal status proposed by Bollen et al. (2006).
It is rather difficult to imagine what an eigenvector score means, but
the following story comes close. A “random journal surfer” randomly
chooses a journal, and an article in this journal, again at random. Then
he or she randomly chooses a reference in this article. Then he or she
moves to this reference’s journal and the whole procedure is repeated