# THREE-DIMENSIONAL LOTKAIAN INFORMETRICS

 DOI https://doi.org/10.1108/S1876-0562(2005)0000005005 Date 20 January 2005 Pages 157-186 Published date 20 January 2005 Author Leo Egghe
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THREE-DIMENSIONAL LOTKAIAN
INFORMETRICS
III.l THREE-DIMENSIONAL INFORMETRICS
In Chapter I we introduced the informetrics theory by describing sources (the "producing"
objects), items (the "produced" objects) and a matching function describing which source(s)
produce(s) which items. In this connection we could talk of two-dimensional informetrics
(sources/items or type/token) which is more than twice one-dimensional informetrics in which
only data with respect to the sources, respectively items are studied (e.g. number of sources,
number of items): in two-dimensional informetrics one also studies and quantifies which
sources produce which items (the matching function) and yields, as a consequence, the
informetric laws (functions).
Given an item set in two-dimensional informetrics we can determine different sets of sources
that produce these items. Indeed e.g. for articles as items one can consider journals (in which
these articles appeared) as sources but one could consider the authors of these articles as
sources as well. Note that, in this chapter, we do not address the problem of multiple
authorship: we will deal with this (intricate) problem in Chapter VI; in the present chapter we
deal with different source sets such as journals and authors producing articles. Reversely (or
in a dual way) we can consider a source set and determine two item sets of produced objects
by these sources. An example is given by articles giving references but also receiving
citations. Note that in the first example an article was an item while in the second example an
article is a source. This leads us automatically to a third case: the case of two source sets and
two item sets but where one source set is equal to one item set. Example: the source set of
journals produces the item set of articles in these journals. This item set is then considered as
our second source set producing a second item set, e.g. of references in these articles. Such
158 Power laws in the information production process: Lotkaian informetrics
systems are the object of study in three-dimensional informetrics. Again also here, three-
dimensional informetrics is more (is a "higher" discipline) than three times a one-dimensional
informetrics theory or than two times a two-dimensional informetrics: the different matching
functions are not independent from each other and, hence, their relations must also be studied.
From the above introduction it is clear that we can determine three types of three-dimensional
informetrics. Let us now describe them in more detail (cf. also Egghe (1989, 1990a) and
Egghe and Rousseau (1990a)).
III.
1.1 The case of two source sets and one item set
This case can be symbolized by Si, S2 the two source sets, I = the item set and depicted as in
Fig.III.l
Fig. III.
1
Three-dimensional informetrics: two source sets and one item
set (—> means: produces).
The most classical example is the one given above: Si = {authors}, S2 = {journals}, I =
{articles},
where we consider the authors and journals as producers of articles. Other
example: I = {articles} produced by Si = {authors} but also by S2 = {research institutes}, i.e.
the research institutes where these authors are employed. Of course, this example could also
be considered in the "linear" way (see Subsection III. 1.3): institutes have authors and authors
have articles. Of the same (double) nature are the examples of Lafouge (1995, 1998): volumes
(of journals) contain articles and these articles are used (e.g. number of (inter)library
requests). Although this example could also be classified in the coming Subsection III. 1.3,
Lafouge also considers the "triangle" form (as in Fig.III.l) with two source-sets and one item-
set. In our notation, his triangle looks like in Fig.III.2 (g,, g2 are the two rank-frequency