View Full Version : The LAW: Alfred Lotka vs Geo. Zipf

James Brody
December 18th, 2005, 12:55 PM
I've had an itch: whenever I see references to "Zipf's Law" I think of Charles Murray's references to Alfred Lotka as a first mover for power laws.

I googled "Alfred Lotka" a few minutes ago and found:

Alfred Lotka
James E. Lovelock, "The Evolution of the Earth" The American scientist Alfred Lotka published a small book, Physical Biology, in 1925. In it he wrote.

It is not so much the organism or the species that evolves, but the entire system, species and environment. The two are inseparable.

As a follower of Alfred Lotka, I want to take his suggestion further and consider evolution as a science that is as much about the rocks and oceans as about the living things that inhabit them. In this view, what evolves is an Earth system that can move gradually for long periods under an ever-warming sun.
Alfred James Lotka (March 2, 1880 - December 5, 1949) was a US mathematician and statistician, most famous for his work in population dynamics.

Born in Lemberg, Austria-Hungary (now L'viv, Ukraine) Lotka's parents were US nationals and he was educated internationally, including a degree at the University of Birmingham, England. In 1935, he married Romola Beattie. They had no children. His varied working life included:

* General Chemical Company
* US Patent Office
* National Bureau of Standards
* Editor of the Scientific American Supplement (1911-1914)
* Staff member at Johns Hopkins University (1922 - 1924)
* Statistician for the Metropolitan Life Insurance Company, New York (1924 until his retirement)

While at Johns Hopkins, Lotka completed his book Elements of Physical Biology (1924) in which he extended the work of Pierre Fran├žois Verhulst and Vito Volterra. His name is most famously associated with the Lotka-Volterra equation of population dynamics.

Lotka, Alfred James (1880 - 1949), USA

Alfred Lotka, chemist, demographer, ecologist and mathematician, was born in Lviv (Lemberg), at that time situated in Austria, now in Ukraine. He came to the United States in 1902 and wrote a number of theoretical articles on chemical oscillations during the early decades of the twentieth century, and authored a book on theoretical biology (1925). He is best known for the predator-prey model he proposed, at the same time but independent from Volterra (the Lotka-Volterra model, still the basis of many models used in the analysis of population dynamics). He then left (academic) science and spent the majority of his working life at an insurance company (Metropolitan Life). In that capacity he became president of the PAA (the Population Association of America).

The article that made him famous as a bibliometrician (avant la lettre) is just a footnote in his oeuvre. He showed that the number of authors with n publications in a bibliography is described by a power law of the form C/na, where C is a constant. The exponent a is often close to 2. Rewriting this equation as a statistical distribution (so that the sum over all n becomes 1), he showed that in the case that a is exactly equal to two, C must be 6/(pi)┬▓, or approximately 0.61. This means that if a bibliography can be described by Lotka's square law, approximately 61% of all authors have contributed just one article to this bibliography.

Lotka A.J. (1926). The frequency distribution of scientific productivity. Journal of the Washington Academy of Sciences, 16: 317-323.
Lotka, A. J. (1925). Elements of physical biology. Williams and Wilkins, Baltimore. [Reprinted in 1956: Elements of mathematical biology. Dover Publications, Inc., New York, New York.
Lotka's Law in OSS Authorship; and, Gauntlet Launches Permalink
By marc on November 20, 2005

My friend Adam, one of the founders of Gauntlet Systems, writes about Lotka's Law, a type of 80/20 distribution, and its applicability to the authorship of open source software:

The 80/20 rule is actually only one of a specific type of numerical relationships known generally as power laws. Much has been written about power laws and their applicability to everything from linguistics to hedge funds. Recently folks have been writing a lot about the power law scaling of web logs and of the "long tail" of web businesses. Way back in 1926, a statistician for MetLife named Alfred Lotka, published a paper in which he observed that the productivity of scientific authors also followed an power-law relationship. Put simply "Lotka's Law" says that a few authors do most of the work, dragging along a long tail of less productive authors.

Power Laws in the Information Production ProcessLotkaian Informetrics
To order this title, and for more information, click here

Leo Egghe, Limsburgs Universiteit Centrum, Belgium

Included in series
Library and Information Science,

This book describes informetric results from the point of view of Lotkaian size-frequency functions, i.e. functions that are decreasing power laws. Explanations and examples of this model are given showing that it is the most important regularity amongst other possible models. This theory is then developed in the framework of IPPs (Information Production Processes) hereby also indicating its relation with e.g. the law of Zipf.

Not sure how Zipf gets more press...