Hyperbolic Distributions in Social Science Zipf’s law and invariants of employment dynamics. Bernd Schmeikal Wiener Institute for Social Science Documentation and Methodology (WISDOM) 10th International Conference of Numerical Analysis and Applied Mathematics, 19-25 September, 2012,
Hyperbolic Distributions in Social ScienceZipf’s law and invariants of employment dynamics
Wiener Institute for Social Science Documentation
and Methodology (WISDOM)
10th International Conference of Numerical Analysis and Applied Mathematics,
19-25 September, 2012,
Kypriotis Hotels and Conference Center,Kos, Greece
we regard social space as not metric
based on a complete partial order
Empirical quantities such as body height, weight of people,
length of nails a. s. o. cluster around a mean value. Considering 4
inch nails, the data set is indeed observed to fit a normal
distribution. But this holds only for a certain type of nail. If we
consider all nails from length, say, 10 mm to 1000 mm, we do not
get a normal distribution, but a power law. Heinz von Foerster
had pointed out in a discussion of Zipf’s law at the Macy meeting
1952: “ … I think the difference in the two kinds of statistics is
that here we are dealing with a number of different kinds,
whereas in the other case in the Gaussian situation, for instance,
- we are dealing with one kind only.”
To comprehend this, let us consider the coastline paradox found
by Lewis Fry Richardson. He observed that the coastline of a
landmass does not have a well-defined length. The length of the
coastline depends on the rulers used to measure it. Since a
landmass can be measured at all scales, from hundreds of
kilometers in size to fractions of a millimeter, there is in principle
no limit to the size of the smallest feature that should not be
measured around. Hence no definite perimeter to the landmass
can be achieved.
When we measure size distributions of labor market shares, we need a measure analogous to the length scale used in the coastline paradox. We are used that frequency-distributions are given by absolute or relative frequencies or briefly ‘number of people’. For example we count the ‘number of scientists’ in a special field of science. These numbers characterize the distribution. But in the RISC-approach the situation is precisely reverse: These numbers now provide us the ‘measure’ for the phenomena we want to measure. We call it the ‘niche-size’. Niche-size is the measure of a labor market share, and, in a way, it parallels the ruler with which we measure the coastline.
The counts that give us the distribution, however, are given by the number of such niches existing in the market. The measure in our RISC-approach is given by the occupation number of a multivariate categorical profile. This is a natural number. So the surveyed metric magnitudes are equal to the sizes of niches in the social space. The social space is a labor market.
We use the labor-service database ‘AMS-BMASK’ and two very large empirical datasets involving fourteen social categorial variables measured in the whole population in order to demonstrate the relevance of the RISC approach and the validity of the statistical model developed.
Selecting gender, age-group, federal state, nation, occupational group, we allow for 8712 combinations which constitute the locations in social space. Each location has a measure given by the size of the niche.
with , and the constant c essentially given by the Riemann zeta function, that is, .
In the year 2001 we obtain and . The following figure shows the approximation.
01/01; Male, 40-44, Stmk, Aut, Building trade, workless 
01/02; Male, 40-44, Stmk, Aut, Building trade, workless, 
01/03; Male, 40-44, Stmk, Aut, Building trade, workless, 
01/04; Male, 60-64, Vienna, Aut, Trade+mot vehicle ind+Rep, workless, 
01/05; Male, 60-64, Vienna, Aut, Trade+mot vehicle ind+Rep, workless, 
01/06; Male, 60-64, Vienna, Aut, Trade+mot vehicle ind+Rep, workless, 
01/07; female, 40-44, Vienna, Aut, Trade+mot vehicle ind+Rep, workless 
01/08; female, 40-44, Vienna, Aut, Trade+mot vehicle ind+Rep, workless 
01/09; female, 35-39, Vienna, Aut, Trade+mot vehicle ind+Rep, workless 
01/10; female, 35-39, Vienna, Aut, Trade+mot vehicle ind+Rep, workless 
01/11; female, 40-44, Vienna, Aut, Trade+mot vehicle ind+Rep, workless 
01/12; Male, 40-44, Stmk, Aut, Building trade, workless 
Zwischenbericht/Interim Report Nr. 47
Regionalentwicklung in Mittel- und Osteuropa: Szenarien für Beschäftigung, Qualifikationen und Migrationsbewegungen
Teil I: Übersichten und Zusammenfassungen
Peter Fleissner, Karl H. Müller (Hrsg.), Bernd Schmeikal, Michael Schreiber et al., Mai 2011
Mag. Richard Fuchsbichler, MBA
Gruppenleiter Gruppe S
Bundesministerium für Arbeit, Soziales und Konsumentenschutz (BMASK)
Sektion VI / Abteilung VI/6
Department of Geography, College of Urban and
Environmental Sciences, Peking University, 100871, Beijing,
China. Email: firstname.lastname@example.org