Page 136 - Becoming Metric Wise

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Publication and Citation Analysis

Proof.:Pick any document d 0 in D and recall that C(d 0 )isthe set of all
21
references of d 0 and C (d 0 ) the set of all documents that cite d 0 .Then C 1
is the set of all documents which either cite d 0 or are cited by d 0 .Bythe
requirement of weak connectedness, C 1 is not empty unless D is equal to
the singleton {d 0 }, in which case the theorem is proved. So we can pro-
ceed and form C 2 . By Theorem 1 we know that d 0 belongs to C 2 .
To show that , N21 C j is equal to D we suppose that some d in D
j50
does not belong to , N21 C j . This assumption leads to a contradiction:
j50
as there is a path, necessarily finite, joining d to d 0 there is a number
j # N 2 1 such that dAC j .

This theorem yields an algorithm for obtaining all the documents in a
given collection, provided the collection is reasonably homogeneous, so
that its citation graph is weakly connected. Moreover, if D is a large com-
puter file, then the algorithm provides a procedure for exploring the core
of a topic (take d 0 to be a core document) and moving further and further
towards the boundaries. This method is known as “cycling.” A mathe-
matical discussion of cycling can be found in (Cummings & Fox, 1973;
Garner, 1967).
The following results provide useful insight into the structure of a
citation

Theorem 3 (Kochen, 1974, p. 21): The average number of references per
document times the number of documents in a collection under investigation is
equal to the average number of citations to a reference item in the collection times
the total number of different references.

Proof.: Let C be the citation matrix of the collection under investiga-
tion. Then c ij 5 1 if document d i cites reference r j and c ij 5 0 if it does
not; note that the index i refers to citing document i, while the index j
refers to cited document j. The columns of this citation matrix contain
only those documents that are cited at least once by the documents in the
collection. On the one hand, if there are n source documents, the average
number of references per document is:
n p !
1 X X
c ij (5.4)
n
i51 j51