Lotkaian Informetrics and applications to social networks

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Lotkaian Informetrics and applications to social
networks

• L. Egghe
• Chief Librarian Hasselt UniversityProfessor
  Antwerp UniversityEditor-in-Chief Journal of
  Informetrics
• leo.egghe_at_uhasselt.be

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1-dimensional informetrics

• authors in a field
• journals in a field
• articles in a field
• references (or citations) in a field
• borrowings in a library
• websites, hosts,
• web citations to a paper
• in- (or out-) links to/from a website
• downloads of an article

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Growth

• Exponential growth
• All new fields grow exponentially
• Otherwise there is S-shaped growth.

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web servers versus time
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2- dimensional informetrics

• authors in a field (sources)
• articles in a field (items)
• indicating which author has written which
  papers
• S Set of sources
• I set of items
• IPP Information Production Process

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Examples of IPPs
S F I
Authors Articles
Journals Articles
Articles Citations (to/from)
Books Borrowings
Words ( types) Use of words in a text ( tokens)
Web sites Hyperlinks (in-/out-)
Web sites Web pages
Cities/villages Inhabitants
Employees Their production
Employees Their salaries

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• size-frequency function
• for n 1,2,3,
• sources with n items
• rank-frequency function
• for r 1,2,3,
• items in the source on rank r
• (sources are ranked in decreasing order of
  number of items they have)

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Continuous model

• Source densities
• Item densities

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Lotkaian Informetrics

• The law of Lotka and the law of Zipf
• Lotka (1926)

. The value is a
turning point in informetrics (see further).
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• Lotkas law is equivalent with Zipfs law

Linguistics Zipfs law in econometrics is called
Paretos law
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Dependence of G on . Existence of a Groos
droop if .
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log-log scale

• decreasing straight line with slope

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Rank-frequency distributions for websites
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The scale-free property

• f scale-free
• such that

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Theorem (i)?(ii)

• f is continuous, decreasing and scale-free
• f is a decreasing power function
• such that
• i.e. Lotkas law

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• Explanation of Lotkas law based on exponential
  growth of sources and items (Naranan (1970)) and
  an interpretation of Lotkaian IPPs as
  self-similar fractals
• (Egghe (2005))
• Fractals and fractal dimension

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• Divide a line piece into 3 equal parts
• ? we need 331 line pieces of this length to
  cover the original line piece
• 3 ? need 331 ? dim1

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• Divide the sides of a square into 3 equal parts ?
  we need 932 squares with this side length to
  cover the original square
• 3 ? need 932 ? dim2
• The same for a cube
• 3 ? need 2733 ? dim3

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Construction of the triadic Koch curve
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• For the triadic Koch curve
• 3 ? need 43D ? dimD
• with

The Koch curve is a proper fractal with fractal
dimension Complexity theory Fractal theory
Mandelbrot
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Naranan (Nature, 1970)

• Theorem
• (i) The number of sources grows exponentially in
  time t
• (ii) The number of items in each source grows
  exponentially in time
• (iii) The growth rate in (ii) is the same for
  every source (ii) and (iii) together imply a
  fixed exponential function
• for the number of items in each source at time
  t.

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• Then this IPP is Lotkaian, i.e. the law of Lotka
  applies if f(p) denotes the number of sources
  with p items, we have
• where

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Egghe (2005) (Book and JASIST)

• (i) The number of line pieces grows
  exponentially in time t, here proportional with
  4t
• (ii),(iii) 1/length of each line piece grows
  exponentially in time t and with the same
  growth rate 3. Hence we have growth proportional
  with 3t.

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• Rephrased in terms of informetrics
• a (Lotkaian) IPP is a self-similar fractal and
  its fractal dimension is given by the logarithm
  of the growth rate of the sources, divided by the
  logarithm of the growth rate of the items.
• (which can be gt or lt 1). Hence, the exponent in
  Lotkas law satisfies the important relation
• This result was earlier seen by Mandelbrot but
  only in the context of (artificial) random texts
  (hence in linguistics).

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Further applications of Lotkaian Informetrics

• Concentration theory (inequality theory) Lorenz
  curves (cf. econometrics).
• Egghe (2005) (Book, Chapter IV).
• Fractional modelling of authorship (case of
  multi-authored articles) determine
• authors with articles
• (fractional counting an author in an
• m-authored paper receives a score ).

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Theoretical and experimental fractional frequency
distributions (case of i4).
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• Dynamics of Lotkaian IPPs, described via
  transformations on the sources and on the items
  includes the description of dynamics of networks.
  
• Relations with 3-dimensional informetrics See
  new journal
• L. Egghe. General evolutionary theory of IPPs
  and applications to the evolution of networks.
  Journal of Informetrics 1(2), 115-122, 2007

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• Item transformation
• Source transformation
• New rank-frequency function

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• Theorem New size-frequency function
• where

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• Case is example of linear 3
  dimensional informetrics
• Sources1 ? Items1 Sources2 ? Items2

• Examples
• Webpages ? hyperlinks ? use of hyperlinks
• Library subject categories ? books
• ? borrowings
• See further.
• Back to the general case.

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• Power law transformations in Lotkaian IPPs

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• Theorem
• is only dependent on b/c due to the
• scale-free nature of Lotkaian systems.

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• Corollary
• With this, one can study the evolution of an IPP,
  e.g. a part of WWW V. Cothey (2007) confirms
  theory except in one case where non-Lotkaian
  evolution is found, probably due to automatic
  creation of web pages (deviation from a social
  network).

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• Further application
• IPPs without low productive sources
• (Egghe and Rousseau (2006))
• Take sources remain but they grow
  in number of items
• Now

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• and (since )
• Evolution decreasing Lotka exponent and no low
  productive sources

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Examples

• Country sizes data from www.gazetteer.de (July
  10, 2005) 237 countries 1.69 (best fit)
• Municipalities in Malta (1997 data) 67
  municipalities 1.12 (best fit)
• Database sizes on the topic fuzzy set theory
  (20 largest databases on this topic) (Hood and
  Wilson (2003))
• 1.09 (best fit)
• Unique documents in databases (20 databases
  above) 1.33 (best fit).

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• Application of Lotkas law to the modelling of
  the cumulative first-citation distribution
• i.e.
• the distribution over time at which an article
  receives its first citation.

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• The time t1 at which an article receives its
  first citation is an important indicator of the
  visibility of research.
• At t1 the article switches its status from
  unused to used.
• t1 is a measure of immediacy but, of course,
  different from the immediacy index (Thomson
  Scientific).

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• The distribution of t1 over a group of articles
  is the topic of the present study. We will study
  the cumulative first-citation distribution
• cumulative fraction of all papers
  that have, at t1, at least 1 citation.

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• Rousseau (1994) uses two different differential
  equations to model two types of graphs a concave
  one and an S-shaped one. These equations are not
  explained and are not linked to any informetric
  distribution.

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• In Egghe (2000), I use only 2 elementary
  informetric tools
• the density function of citations to an
  article, t time after its publication
  (exponential, ),
• the density function of the number of
  papers with A citations in total (Lotka,
  ), (only ever cited papers
  are used here).

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• Normalizing to distributions
• becomes for an article
  with
• A citations in total
• becomes but we will use
• the fraction of ever cited articles, in
  order to include also the never cited articles.

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• Theorem
• concave if
• S-shaped if
• , hence explaining both shapes in one model.
• Note the turning point of .

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• Proof A first citation is received if
• ()
• ? Cumulative fraction of all articles that are
  already cited at time t1
• ()
• ? () into () yields

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Motylev (1981)
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• fit

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Rousseau (1994)
JACS to JACS data of Rousseau Time-unit 2
weeks, 4-year period

• fit