Page 140 - Becoming Metric Wise

131
Publication and Citation Analysis

following proposition is just a re-interpretation (in a different context) of
the previous one.

Proposition 2: The number of articles of a given author a i is
n t
P
ð
ð
j51 w ij 5 W UÞ 5 W W Þ , where U is the column vector consist-
ii
i
ing completely of 1’s.
The proof is exactly the same as that of Proposition 1. The only dif-
ference is that C is replaced by W. Clearly similar propositions hold with
other matrices. This shows the power of mathematics. One proof holds
in all similar circumstances; only the interpretation is different.
Proposition 3: The number of citations received by a given paper r j is
P m t t
c
i51 ij 5 U Cð Þ 5 C CÞ jj
ð
j
Similarly we have Proposition 4.
The number of coauthors of paper j is
P m t t
w ij 5 U WÞ 5 W WÞ
ð
ð
i51 j jj
Again the same calculations prove Propositions 3 and 4. We prove
Proposition 4 as an example.
Proof: The matrix W consists of zeros and ones, ones if the corre-
sponding cell is occupied and zero otherwise. Keeping the column j fixed
P m
i51 w ij is just the number of ones in the j-th column. This is the num-
ber of times paper r j has an author, or the total number of authors of
paper r j .
t t
Now U W is an (n,1) matrix, i.e., a column vector. U Wð Þ is
j
the j-th element of this column vector. It is equal to:
m
m P
m
t
t
P
P
ð U WÞ 5 i51 ð U Þw ij 5 1:w ij 5 w ij .
j
i
i51 i51
t
t
Similarly ð W WÞ 5 P m ð W Þ : WÞ 5 P m w ij w ij 5 P m w ij ,
ð
jj i51 ji ij i51 i51
2
where, again, this last equality follows from the facts that 1 5 1 and
2
0 5 0.
5.10 BIBLIOGRAPHIC COUPLING AND COCITATION
ANALYSIS
5.10.1 Bibliographic Coupling
Bibliographic coupling and cocitation are two notions used to describe
mutual relations in a citation network. Until now we studied relations
that can be described as Cit(A;B), i.e., A cites B. In this section we move