Laboratoire Ursidoc
Université Claude Bernard Lyon 1
43 Boulevard du 11
novembre 1918 69622 Villeurbanne Cedex, France
Lafouge@univ-lyon1.fr
Agnieszka
Smolczewka
Laboratoire Ursidoc
Université Claude Bernard Lyon 1
43 Boulevard du 11
novembre 1918 69622 Villeurbanne Cedex, France
Abstract
Statistical distributions in the production or utilization of information are most often studied in the framework of Lotkaian informetrics. In this article, we show that an Information Production Process (IPP), traditionally characterized by Lotkaian distributions, can be fruitfully studied using the effort function, a concept introduced in an earlier article to define an Exponential Informetric Process. We thus propose replacing the concept of Lotkaian distribution by the logarithmic effort function. In particular, we show that an effort function defines an Exponential Informetric process if its asymptotic behavior is equivalent to the logarithmic function with , which is the effort function of a Lotkaian distribution.
1. Introduction
Statistical
regularities observed in the production or use of information have been studied
for a long time in informetric processes. Today, they are again very topical,
as is testified by the many articles. They are characterized by phenomena of
invariance of scale during research into the traffic on Internet (Aby & al.,
2004), (Barabasi & al., 2000). They are also observed when the topology of
the Web is studied (Bilke & al., 2001) or when counting the frequencies of
the number of pages or the number of degrees entering or leaving the Web pages
in a collection of sites (Prime Claverie & al., 2005). Their most current
mathematical formulation is that of an inverse power function, usually called a
Lotkaian informetric distribution. A
recent book from Egghe (Egghe, 2005) proposes a mathematical approach to the
framework of Lotkaian informetrics, illustrated by several examples.
In this article, we continue a study begun previously
(Lafouge & Prime Claverie, 2005) where we defined the Exponential Informetric
Process by introducing the concept of effort function. We are studying here
informetric processes while drawing on traditional mathematical formulation in
continuous mode. Mathematically, we represent the effort function by the
logarithmic function, which is related to the effort function that appears in
the law of Lotka (Lotka, 1926).
2. Information Production
Process and effort function
Statistical
distributions in the production or utilization of information, such as the law
of Lotka (Lotka, 1926) - production of articles by researchers in a scientific
community - generally fit into simple unidimensional models. These models can
be represented by the diagram of Figure 1, introduced into informetric systems
by Leo Egghe (Egghe, 1990) and called "Information Production
Process" (IPP). An IPP is a triplet made up of a bibliographical source, a
production function, and all the elements (items) produced.
Figure 1: Schematic representation of an Information
Production Process
In (Lafouge & Prime Claverie, 2005) we assume that an item produced
requires a certain amount of effort and therefore we define the informetric
process by introducing the effort function (see Figure 2). We use the size
frequency form, and denote as the frequency distribution where represents the number
of sources that have produced items (maximum number of
items produced). The effort functiondenotes the amount of effort from a source needed to produce i items . The amount of effort, denoted as , produced by an IPP is:
If is the identity
function , the amount of effort produced by the process is simply
equal to the number of items produced. Since the production and the effort
(function) appear to be logically connected, we use both to define the
Exponential Informetric Process.
Figure 2: Schematic representation of an informetric
process using the effort function
3. Exponential Informetric Process
In the article (Lafouge & Prime Claverie, 2005) we
define an Exponential Informetric Process in terms of an exponential
density and an effort function where the average quantity supplied by the
sources to produce all the items is finite. More precisely, we define a set of
functions denoted:
EF =
We call effort function
an element of. Let be a number greater
than 1, ,
we call Exponential
Informetric Process the following density function :
( constant of normalization) [1],
where
is finite [2].
corresponds to the average of effort produced by the density
process . The results (b) and (c) that follow, explain the
relationship between average of effort and entropy.
3.1. Entropy and effort
The Maximum Entropy Principle (MEP) maximizes the entropy
subject to the constraint that the effort remains constant, whereas the
Principle of Least Effort (PLE) minimizes the effort subject to the constraint
that the entropy remains constant (Egghe, 2005).
Assuming an effort function and
a number greater than 1, we show in (Lafouge & Prime
Claverie, 2005) that these conditions imply the following properties:
(a) is a function with decreasing density over the interval.
(b) The two principles, maximum entropy and
least effort are verified simultaneously.
(c) If
and describe the average
information content and effort produced by the process, we have the following
proportional relationship: .
Note: Here
we will use , where is the Euler number,
and. All the results are valid for a logarithmic function in any
base.
Also, in the following,
for we denote as the associated density function (we suppose ). is an Exponential Informetric Process if condition [2] is
verified, that is if the average effort, is finite.
Note 1
More generally, we can easily show that if defines an Exponential Informetric Process and that is a limited positive
function, then also defines an
Exponential Informetric Process, provided that h is an increasing function on the interval, where . This note enables us to
envisage building a multitude of effort functions starting from an Exponential
Informetric Process (see Example c) below).
Note 2
It is important to note that there exist effort functions
for which is a density function
but the condition [2] is not satisfied.
The reader will see
that, for the following effort function:
we have
and that is infinite.
Proposition 1
Assume and two effort functions,
so that is greater or equal
than on the interval where. If is an Exponential Informetric Process the same holds for .
Proof
Given that and are increasing
functions not majorized such that , there is a positive number greater than 1 such
that. Knowing that the function is decreasing on the interval we can write:
We can therefore
conclude that if is an Exponential Informetric
Process the same holds for .
□
Considering Note 2 and the result of Proposition 1, it would be interesting
to define an Exponential Informetric Process with a different condition from
condition [2]. This condition and related results obtained by means of Proposition 1, will be presented in the
following.
4. Characterization of an
Exponential Informetric Process
As mentioned before, we seek to define an adequate
condition that will allow us to say if any effort function defines an
Exponential Informetric Process. To do this, we will compare the asymptotic
behavior of an effort function with the effort functions of a Lotkaian
distribution.
4.1. Lotkaian distribution
We saw in the introduction that the most current
formulation for modeling many phenomena related to the use or production of information
is that of Lotka, given here in continuous form:
[3]
The properties of these
hyperbolic distributions or inverse power laws have been widely studied (Egghe,
2005). It seems essential to us to be able to compare these functions with an
exponential density, defined by an effort function.
We easily go from [3]
to [1] taking as effort function , where. The effort function of an inverse power function is a function
proportional to the logarithmic function. This characteristic clarifies the
principle of the least effort. This result is implicit in the article of
Yablonsky (Yablonsky, 1981), where he shows the relationship between the
principle of the maximum entropy and an inverse power function. Also, it is
known that such a distribution (Lafouge & Michel, 2001) has an entropy
equal to and thus a quantity of
effort equal to . An inverse power function thus defines an Exponential
Informetric Process (noted) if is strictly positive.
For less than or equal to
0, is not an Exponential
Informetric Process and neither is a density function.
4.2. Effort function and Exponential Informetric
Process
4.2.1. Characterization by logarithmic effort function
In the following, we will characterize the Exponential
Informetric Process by the effort function of a Lotkaian distribution.
Theorem 1
Let be an effort function
so that exists[1], and
verifies the condition
[4],
where C is a finite
number greater than one or infinite. Then defines an Exponential
Informetric Process.
Proof
a) is finite
In order to define
an Exponential Informetric Process it suffices to show that condition [2] is
verified. Since , then it exists a finite number such that Let and , then
According to the
preceding, we know that defines an Exponential Informetric Process.
Condition [4] implies:
so that
so that .
Choosing , we then have . Since is an increasing function not majorized we can choose A so that .
being an Exponential Informetric Process, the preceding
proposition allows us to say that defines an Exponential
Informetric Process.
b) is infinite
The relationship [4]
allows us to write:
so that.
Proof follows, as for
the finite case.
□
In other words, an
effort function defines an informetric process if its asymptotic behavior is
equivalent to the logarithmic function with.
Examples
a) Exponential case
This case corresponds
to a linear effort function:
where:
We find the well-known case of an exponential distribution
:
b) Mixed case
This case corresponds
to an effort function that is the sum of a linear type function and a
logarithmic type function. The effort function is:
In this case .
The exponential process corresponding to the density
function is :
,
which corresponds to the distribution of Erlang. If is real and
non-integer, we recognize the gamma distribution.
c) General inverse power law
The following example
is an inverse power law modified for low frequencies. This case corresponds to
an effort function of type:
This function is increasing for .
In this case, we have . We can then conclude.
The associated density function is:
,
where is a standard
constant. This distribution, used for example in (Naranan, 1992), allows us to
adjust statistical distributions on the vocabulary of various language corpora.
In general, calculations give us close to 2 and less than 1. This
distribution is one of many possible generalizations of the law of Zipf (Shan,
2005). According to the sign of the constant, the effort function will have a different effect for low
frequency values, which are known to have an important influence. In this case,
the adjustment will be done via a multi-linear regression after transforming
the data on a logarithmic scale.
Note 1 allows us to consider many other functions of this type.
d) Other examples
Among many other
possible examples we have, for instance, the normal Log law with an effort
function of type:
Every effort function
of type:
also defines an Exponential Informetric Process.
We can show that for
these functions, the quantity of effort is finite and is equal to:
We can also quote the
distribution of Weibul here. This distribution, like the normal Log
distribution, can be increasing then decreasing. This is why we require an effort
function to increase only on the interval where and not necessarily on
the interval.
□
Regrettably, the
preceding theorem is not a necessary and sufficient condition. In fact, when is equal to 1, the two following cases are possible:
- if we know that is not an informetric process
- if the reader can show
that
Nevertheless, we can
provide a necessary condition, presented in Theorem
2.
Theorem 2
Assume g is an effort
function so that the exists and verifies:
[5]
where C is a positive number less than 1, then
is not recognized as
an Exponential Informetric Process since its average of produced effort is
infinite.
Proof
Since , then it exists a finite number verifying Let and , then
Since , we know that is not an Exponential Informetric Process.
Condition [5] implies:
so that . Therefore .
Since is an increasing function not majorized we can choose A so that:
.
Since the quantity of
effort is infinite, we can
conclude using the same argument as in Proposition
1.
□
4.2.2.
Characterization
by generic effort function
The two previous
theorems can be generalized replacing the logarithmic function by an effort
function having a finite (Theorem 3)
or infinite (Theorem 4) quantity of
effort.
Theorem 3
Assume is an effort function
and an Exponential Informetric
Process so that exists and verifies
the condition:
with , then is also an Exponential
Informetric Process.
Proof
The same as in the Theorem 1, using the argument:
.
□
Like Theorem 3, which generalizes the result
of Theorem 1, the following theorem
generalizes the result of Theorem 2.
Theorem 4
Assuming is an effort function and an effort function
with an infinite quantity of effort and verifying the condition:
exists and verifies with ,
then is not an Exponential
Informetric Process, since its quantity of effort is infinite.
The proof is the same
as in Theorem 3, using the argument
that the quantity of effort is, in this case,
infinite.
4.3.
Moment and Exponential Informetric Process
It is a known fact that a Lotkaian distribution has only
moments of order for strictly less than (where is the exponent of the inverse power law). An Exponential
Informetric Process may have, or not have, moments of order n. We will give a sufficient condition
so that an Exponential Informetric Process has a moment of order n. In order to do this, we will compare
the asymptotic behavior of an effort function with an Exponential Informetric
Process, whose effort function is the power function , where n is an
integer greater than 1.
Theorem 5
Let be an integer greater
than 1 and an effort function so
that exists and verifies
the condition:
where C is a finite number greater than one or
infinite, then defines an Exponential
Informetric Process characterized by moments of order.
Proof
According to Theorem 1, is an Exponential
Informetric Process. Let us show that it has moments of order n:
a) is finite
so that , we have ,
therefore for , we have ,
therefore .
We can then conclude
because is an Exponential
Informetric Process.
b) is infinite
The proof is identical.
□
5.
Conclusion
An IPP (Information
Production Process) is defined by sources, a production function and the
elements produced. Given that each element produced requires a certain quantity
of effort, it is justified to approach an IPP as much by its effort function as
by its distribution. In an Exponential Informetric Process, both are closely
linked.
In informetrics, the
traditional cases of IPP are generally characterized by Lotkaian distributions.
Given that it is justified to approach an IPP by the effort function, we here
propose replacing the concept of Lotkaian distribution by the logarithmic
effort function. The results presented in this article, in particular Theorem 1 and Example c), show that if a distribution of effort is equivalent to
the logarithmic function with , we have an Exponential Informetric Process.
Acknowledgements
The authors wish to
thank A. Agouzal,
References
Aby, P.,
Flandrin, P., Hohn, N., & Veitch, D. (2004). Invariance of scale in the
Internet, published in "Mesures de l'Internet" under the direction of
Eric Guichard, Les Canadiens en
Barabasi, A.L., Albert, R. & Jeong, H. (2000). Scale-free characteristics of random networks: the topology of the world-wide-web. Physica A, 311, 590-614.
Bilke, S. & Peterson, C. (2001). Topological properties of citation and metabolic networks. Physical Reviews E, 6403(3), 76-80.
Egghe, L. (1990). On the duality of informetric systems with application to the empirical law. Journal of Information Science, 16, 17-27.
Egghe, L. (2005). Power laws in the information production process: Lotkaian Informetrics. Elsevier.
Lafouge, T., & Prime Claverie, C. (2005). Production and use of information. Characterization of informetric distributions using effort function and density function. Information Processing and Management, Vol 41, 1387-1394.
Lafouge, T., & Michel, C. (2001). Links between information construction and information gain. Entropy and distributions. In Journal of Information Science, 27(1), pp 39-49.
Lotka, A.J.
(1926). The frequency distribution of scientific productivity. Journal of the
Naranan, S. (1992). Statistical laws in information science, language and system of natural numbers: some striking similarities. Journal of scientific and industrial Research 51(8-9), 736-755.
Prime Claverie, C., Beigbeder, M., & Lafouge, T. (2005). Limits and feasibility of cositation method on the web an experiment on the French speaking Web; In Proceedings of the 10 th international conference of scientometrics and informetrics, p78-86, July 2005, University of Stockholm, Sweden
Shan, S. (2005). On the generalized Zipf distribution. Part 1. Information Processing and Management, Vol 41, 1369-1386.
Yablonsky, A.L. (1981). On fundamental regularities of the distribution of scientific productivity. Scientometrics 2(1), 3-34.