Fractanomics - The Issue of Scale in the Network Economy

draft version 0.3; November 12, 1997

1. Introduction

Let's face it, most success seem to migrate to those people or companies who already are very popular. In Holland we have a saying that 'the devil shits on the largest shitheap', meaning that the rich get richer all the time. The Bible also mentions the inequality of the way people use their 'talents' over the years.
A more recent example of this phenomenon is the very wide scale of visitor frequency of WWW pages [1]. Some popular pages have millions (and growing) of hits a day and others have only a few. Success seems to attract success, which can result in growth. Whatever the feelings of unfairness and fears of market dominance these inequalities may raise, it looks very interesting to take an objective closer look at what mechanisms may be behind this phenomenon.

2. Problem definition

A new form of market structure is spreading all over the world. This chain of flexible human transaction networks is called the Network Economy. It is acting commercially on a more and more world-wide scale and at the same time it fragments the old social structures into tribes [2]. This new dynamic development, which is very much fuelled by the Internet, results in symptoms that seem to contradict:

So the question is:

3. Analysis

In 1949 the linguist G.K. Zipf published a book [3] in which he empirically showed that systems that involve life do exhibit certain statistical characteristics that fit a simple power law. For instance, the frequency of occurances of written (ASCII) characters, or words in a book, if charted by rank #n in decreasing popularity, very well fit a curve if 1/n.
For example, in Dutch 'de' appears most often, 'en' appears half as often and 'een' about one third as much, etcetera. The first few elements score very high. There is a medium number of elements with middle-of-the-road scores, and a huge number of elements that nearly never occur. This pattern is the basis for the efficiency of for instance Morse coding, where the more frequent characters are given a shorter code. Another example is disk caching or mirroring where the most often accessed data is kept in memory for expected re-use.
This striking inequality of the ranking distribution graphs is also know as 'the 80/20 Law' or Bradford's Law and it appears in the quality graphs of the Pareto method and in the spatial distribution graphs of population density of for instance Weber. More formally: Zipf's Law states that probability P(r) of occurance of an element with rank r is

For words in the Dutch language c= 0.13 gives the best fit.

Very many things in society exibit this empirical regularity, like the size of cities, the turnover of insurance companies, networknode datatraffic, airport traffic.
More recently, measurements show that access to WWW documents very well fit a Zipf law [4], [5]. It must be stressed however that Zipf's law is not a statistical distribution of frequency versus a parameters like the Gauss curve, but a ranking distribution of elements that do dynamically change place in rank continuously. Insurance companies for instance merge, but sometimes they break up into parts that are sold off or outsourced. Billionaires rated on the Forbes List are on the way up or they are on the way down on that list.


a. if this Zipf rank distribution of success appears objectively to be a fact of life,

b. what mechanism is behind the inequality of popularity? Can we explain how it works like it does?

c. Some can judge the graphs to be an undesirable state of affairs. Can they influence its outcome in their own interest or, for instance try to correct its unfairness for the sake of the common good or to protect the less fortunate?

It is unfortunate that Zipf and others did not make the above distinctions a/b/c and have made very politically incorrect claims on the basis of a. to conclude for c. that elite structures are what we should desire. The position of this paper is that if a. is true we should first try to find more knowledge about b. Views on c. are considered premature here.
Zipf tried in his book to derive his law on the basis of a 'tool analogy' and a principle stated by him that Nature is lazy, in that it wants to save energy. Nature, according to Zipf, favours the efficient use of energy resources by choosing the (Lagrangian?) route of least effort. In simple terms: people will lay out (rank) their tools in such a way that the most often used tool is nearest at hand. Because of this very reason they will also choose those tools more often, because they are nearest to the worker. Besides this virtuous circle the derivation contained circular arguments and can not be considered a rigourous proof or even proper explanation.

Recently scientists from different fields have shown that Zipf's Law Rankdistribution in certain cases appears by the simple process of repeated random selection from an equi-probable group of items [6]. So a large group of monkeys who press buttons choosing them at random would over time produce the Zipf curve. Nothing strange or mystifying about it?

Well, some things remain unclarified by this simple explanation. In the high ranking part of most graphs as well as in the low ranks the curves of measurements often do not fit. To be more specific, in some cases the first few very popular items tend to be less different in size than the Zipf formula would dictate. In other cases the inequality is even more steep than Zifp's law. In both kinds of differences the other end (the very unpopulars) of the graph also does not fit properly! In 1951 the now famous mathematician Benoit Mandelbrot derived Zipf's law as a special case of a more broadly valid formula:

In the case of the Dutch words, see above, the best fit is for c=0.2; p=2 and B = 1.09.

Mandelbrot's derivation of Zipf's law dates to 1951 and is sketched briefly on pages 344 to 347 of his famous book [7] about fractals. Prof. Mandelbrot wrote to the author that a whole chapter on Zipf's law, including a more clarified formal derivation, is part of his new book [8]. Not only does the Mandelbrot Law (or should we say Zipf++) fit better to the data it also gives a clue about what mechanisms might be at work.
The proof is based on the assumption that the living system, in which the choices can be made leading to a broad ranking of popularity, exhibits a fractal structure. This means that in any scale in the system we can find the same kind of patterns repeated. Popular example is a tree for which the branching pattern seems repeated into the leafs of the tree. Other examples are clouds, the road infrastructure and coastlines/ beaches/ heaps of sand and water. On any scale these are 'self similar'.

So if systems that exhibit fractal patterns can be proven to produce the Zipf++ ranking distribution, and in the new Network Economy the popularity of organizations does fit the Zipf++ graphs we can speculate that the modern companies behave in a fractal way and have certain structures and behaviours that are similar to other systems that are fractal organisms.

That 'fractal company theory' is exactly what was put forward recently by a number of organizational specialists like Warnecke [9] and Kühnle [10],[11]. They talk about revolutionary and very urgently needed reorganization of German companies which has to go much further than the shocking 'reengineering'. Warnecke and Kühnle introduced the terms :

Instead of the old and stifling slow Taylorized companies with centralised planning and control, specialisation and bureaucracy; the German professors propose new structures which exhibit the properties of Fractal systems: networked self-organization, self-similarity and dynamic flexibility. Such structure would be much better to cope openly with complexity and unplannable turbulence.

4. Synthesis and recommendation

This paper proposes to describe the repeating pattern in fractal companies as 'networked objects', as in 'object oriënted programming'. From the level of the global network economy to the fractal companies and their fracteams all the way to the fractworkers the same organization model is appearing: an object with an outer skin containing all necessary component objects to perform a series of functions which are performed in communication/dialogue with its environment. This model is also repeated on the ICT plains of business organization, computer systems as well as in digital switching and transmission. We might even say that such structure of 'enclosed & linked' objects is valid down from biological cells all the way up to (and beyond?) the planet.

The word fractal is derived from the Latin word 'fractum' meaning fragment. And globalisation (upsizing) as well as fragmentation (downsizing) was the effect that was described by Castells in [1] as resulting from the new Network Economy. So if a wide scale fractal structure is behind that development which does grow uncontrolled and in an organic life-like way, that might be a valid explanation, which at the same time can answer the question about the best possible size for a fractal company, see above. But before we propose an answer, we introduce the following Roadmap Metaphor here:

So if a road is an example of a 'connected fractal object' then the answer to the question of chapter 2 is : In fractanomics there is no best size! Any size of fractal company can be succesful depending on :

  1. Value generators: what variety of knowledge skills you have on board to add value to the outside world?
  2. Well Connectedness: the worldwide reach and quality of your network connections: can they reach you?
  3. Initiative and learning: the first one who starts a contagious/magnetlike service, however small, can be wildly succesful and will stay ahead of any copyist service, if and only if the starter learns and keeps learning together with his clients/members of a virtual community.

Furthermore, big companies also will internally reorganize to consist of a network of fracteams. The fractal pattern will appear inside as well as outside companies, involving clients and suppliers, blurring present borders. The prevalent model will be: fracteam units that are "independant & well connected", having all possible sizes and which are interconnected into a fractal patterned network structure. Such network of interconnected and communicating fracteams, each with knowledge and skills integrated on-board, can handle dynamic complexity much better than the old structure of Taylorized fragmentation and simplification that was due to the archaic tools of communication then available. The Internet digital communication networks make all the difference!

In the open society and global open marketplace where people are free to choose, such fracteams will show up in the charts that fit the Zipf++ Law, not by their size but by their rate of growth: a short doubling time and a large candidate client population. And if the emerging structure behind this new economy is indeed fractal, then the components have a lot in common. The fate of each part, however small, is of consequence to the organism as a whole. The large must have concern over the small and vice versa because it is part of them too. 'Whatever you do to the web you do onto yourselve'. Together we can cope with the complexity and turbulence of Reality. Not by installing more computerpower at a vulnerable central planning and control center.
Ross Ashby's law of requisite variety shows that that strategy is fruitless even with supercomputers. The way ahead is to grow a worldwide fractal network of intelligent learning units that resembles the organization of a brain. Maybe this Network is what Paul Baran had in mind when he laid down the fundamental design principles of Internet many years ago.

Yes, there is a certain amount of wishful thinking/visioning in this paper. The author likes to feel that there must be more to life than us being monkeys that click on a choice of Windows buttons in a purely random way. Fractal patterns are a sign of life. And ...they do emerge.

[1] Nielsen, Jacob; Do Websites Have Increasing Returns?, april 15, 1997 04b.html

[2] Castells, Manuel; 'The Information Age: Economy, Society and Culture', Vol.I The Rise of the Network Society (1996), Vol.II The Power of Identity (1997), Vol III End of Millenium (to be published soon); Blackwell Publishers, Oxford.

[3] Zipf, George K.; 'Human Behaviour and the Principle of Least-Effort', Addison-Wesley, Cambridge MA, 1949

[4] Nielsen, Jacob; Zipf Curves and Website Popularity, April 15, 1997 f.html

[5] Cunha, Bestavros, Covella; Characteristics of WWW Client-based Traces, July 18, 1995; l

[6] Turner, G.R.; Relationship between Vocabulary, Text Lenght and Zipf's Law, Sept.1997. m/~g.r.turner/ZipfDoc.htm

[7] Mandelbrot, Benoit B.; "The Fractal Geometry of Nature" ; WH Freeman 1982.

[8] Mandelbrot, Benoit B.; "Fractals and Scaling in Finance: Discontinuity and Concentration"; Springer-Verlag TELOS; Expected publication date: November 1, 1997 ISBN: 0387983635 /exec/obidos/ISBN=0387983635/e

[9] Warnecke, Hans-J.; 'Revolution der Unternehmenskultur -Das Fraktale Unternehmen'; Springer-Verlag Berlin Heidelberg 1993, ISBN 3-540-57196-5 Professor Warnecke ist Leiter des Fraunhofer-Instituts für Produktionstechnik und Automatisierung und seit Oktober 1993 Präsident der Fraunhofer-Gesellschaft.

[10] Kühnle, Hermann; in: 'Aufbruch zum Fraktalen Unternehmen - Praxisbeispiele für neues Denken und Handeln', Ed: Warnecke; Springer-Verlag Berlin Heidelberg 1995, ISBN 3-540-58668-7

[11] Kühnle, Hermann; Paradigmenwechsel in der Production - Die Fraktale Fabrik; Jan.7, 1997 http://iaf

Jaap van Till
TU Delft, Stratix Consulting Group B.V. and cofounder of the ISOC-NL Chapter