Page 137 - Becoming Metric Wise

128 Becoming Metric-Wise

P p
c
Here p denotes the total number of different references and j51 ij is
the number of references contained in d i . The citation matrix is an (n,p)
matrix. Then the average number of references per document times the
number of documents is equal to:

n p !
X X
c ij (5.5)
i51 j51
which is the total number of 1’s in the citation matrix. On the other
hand, we see that the average number of citations to a document in the
collection is

!
p n
1 X X
c ij (5.6)
p
j51 i51
n
X
where c ij is the number of citations received by the j-th reference
i51
from documents in the collection. Then, the average number of received
citations times the total number of different references is also equal to the
number of 1’s in the citation matrix. This proves the theorem.
An application. Assume that you have a (20,120) citation matrix and
you know that the average number of references per document is 10.
How many documents are there in this collection? And what is the
average number of received citations for the references in this collection?
From the dimensions of the citation matrix we know that the
collection contains 20 documents (and that there are 120 different
references).
The average number of citations is then equal to (1/120) 3
(10 3 20) 5 5/3 1.67.

5.9.3 The Publication and Citation Process Described by
Matrices

Although this section is useful for performing or checking some calcula-
tions, it may safely be skipped by those readers who are not really familiar
with matrix calculations. In Subsection 5.9.1 we have shown how a citation
matrix is written and how it is related to a citation network. In this section
we will show how notions such as the number of articles, number of refer-
ences in a given paper or the number of references common to two articles
can be obtained (automatically) from such a matrix. Similarly, starting from