LOTKAIAN FRACTAL COMPLEXITY THEORY

 Pages 231-246 Date 20 January 2005 DOI https://doi.org/10.1108/S1876-0562(2005)0000005007 Published date 20 January 2005 Author Leo Egghe
V
LOTKAIAN FRACTAL COMPLEXITY
THEORY
V.I INTRODUCTION
The most natural and exact way in which complexity of a system of relations can be described
is by indicating its dimension in space. As an example we can consider a citation analysis
study of how n journals are citing another k journals (possibly, journals can appear in both
sets;
both sets can even be equal in which case n = k but this is not a requirement). For each
journal in the first set we can look at all the references that appear in the articles in this journal
and in this way we can count the number of times one finds a reference to an article that was
published in a journal that belongs to the second set. We hence obtain n vectors with k
coordinates, hence an n x k matrix in which entry ay (ie
{l,...,n},
je {l,...,k}) denotes the
number of references, appearing in articles of journal i, to articles of journal j. Such a
situation can be considered as a cloud of n points that belong to k-dimensional space Rk (R
denotes the real numbers). We here have the problem of visualizing (or at least describing)
this k-dimensional cloud of points. We will not describe the techniques of multivariate
statistics that can be used here in order to reach this goal: one uses dimension-reducing
techniques such as principal components analysis, cluster analysis or multidimensional
scaling in order to visualise the cloud of points in two dimensions, i.e. on a computer screen
or a sheet of paper - see Egghe and Rousseau (1990a) for an extensive account on this,
including several examples from informetrics.
The purpose of the present chapter is not to reduce the dimension of such a situation but
rather to describe it mathematically. We will also limit our attention to the description of the
dimension of IPPs, henceforth called the complexity of IPPs. Our main goal in this book, of
course, will be the study of complexity of Lotkaian IPPs, i.e. where we have a size-frequency
232 Power laws
in the
information production process: Lotkaian informetrics
function
f
that
is of the
power
law
type (11.27).
So,
here,
we do not
study
(as in the
example
above) which source (journal) cites which source (journal)
but the
situation
"how
many
sources
(e.g.
journals) have
how
many items
(e.g.
citations)"
and
this from
the
viewpoint
of
the "dimensionality"
of the
situation.
In
other words
we
will describe (Lotkaian) IPPs
as
fractals.
A
fractal
is a
subset
of
k-dimensional space
lRk (k = 1,2,3,4,...) but,
dependent
on its
shape,
it
does
not
necessarily incorporate
the
full k-dimensionality
of Rk. A
simple example
is
a
straight line
in Rk
which
is a
one-dimensional subset
of Rk or a
plane
in Rk
which
is a
two-dimensional subset
of Rk.
In
the
next section
we
will study dimensionality
of
general subsets
of Rk: e.g.
explain
why a
straight line
has
dimension
1 and a
plane dimension
2 but we
will also study "strange" subsets
of
Mk for
which
we
have
to
conclude that their dimension
is not an
entire number
but a
general number
in K+ (the
positive real numbers): such sets
are
called proper fractals.
The
reason
for
this study
is
their interpretation
in
terms
of
IPPs:
it
will
be
seen
(in
Section
V.3)
that IPPs
can be
interpreted
as
proper fractals. Lotkaian IPPs will
be
characterized
as
special
self-similar fractals,
i.e.
fractals (subsets
of Rk)
which
are
composed
of a
certain number
of
identical copies
of
themselves
(up to a
reduced scale). Therefore
our
study,
in the
next
section,
of
fractals will focus mainly
on
such self-similar fractals
and we
will
see
that,
for
such sets,
it is
easy
to
calculate their fractal dimension (also introduced
in the
next section).
As said,
in
Section
V.3, we
will interpret IPPs
as
fractals
and
show
the
special self-similarity
property
of
Lotkaian IPPs interpreted
as
fractals.
It is now
clear that
the
Lotka exponent
a
must play
a
central role
in
this:
we
will show that
a -1 is the
fractal dimension
of the self-
similar fractal associated with this Lotkaian
IPP.
This, again, shows
the
importance
of
power
laws.
V.2 ELEMENTS
OF
FRACTAL THEORY
In this section
we
will, briefly, review
the
most important aspects
of
fractal theory that
we
will need further
on. For a
more detailed account
on
these matters
we
refer
the
to
Feder
(1988)
or
Falconer (1990)
or to
Mandelbrot (1977a),
the
founding father
of
fractal theory
and
also
the one
that formulated
the law of
Mandelbrot
- see
Chapters
I and II; in
fact
his