eFirst back to homepage | back | next | Last
bordalier institutecontact:  winiwarter@bordalierinstitute.com


Research scope :   complex systems / neural networks &  evolution


"I think the next century (21st) will be the century of complexity." Stephen Hawking

brainUniverse

"Life and mind have a common abstract pattern or set of basic organizational properties. The functional properties characteristic of mind are an enriched version of the functional properties that are fundamental to life in general. Mind is literally life-like. "

                   
                                             Godfrey-Smith, P. (1996). Complexity and the Function of Mind in Nature. Cambridge: Cambridge University Press.

"Mind is literally life-like. The Universe and Life are literally mind-like. "

                   
                                             Peter Winiwarter (2008). Network Nature. www.bordalierinstitute.com



Contents : it may take several minutes to download the numerous illustrations.


  1. Keywords math
  2. Abstract

  3. evolution : the big picture  a synopsis with links

  4. empirical observations

"It is an interesting possibility that the power laws followed by so many different kinds of systems might be the result of downward constraintes exerted by encompassing supersystems."

Stanley N. Salthe, Entropy 2004, 6, 335

  1. empirical observations :  astrophysics

  2. empirical observations :  geophysics, geology, geoscience, meteorology

  3. empirical observations :  bio-chem, biology, plant, animal, ecosystems

  4. empirical observations :  economics,  social systems

  5. empirical observations :  religion, sociology

  6. empirical observations :  linguistics

  7. empirical observations :  mechanical systems, communication systems, computer tech

  8. empirical observations :  science, education, innovation

  9. empirical observations :  world wide web

  10. empirical observations :  brain, neural networks

  11.  theoretical approaches    a synopsis with links

    Zipf's principle of least effort,

    Pareto efficiency Pareto frontier
    ,

    Holland Gell-Mann complex adaptive systems

    Preferential attachment

    Odum's maximum power principle*

    That theory, as it is expressed by the maximum power principle, addresses the empirical question of why systems of any type or size organize themselves into the patterns observed. Such a question assumes that physical laws govern system function. It does not assume, for example, that the system comprising economic production is driven by consumers; rather that the whole cycle of production-consumption is structured and driven by physical laws.

    West's min-max principle for hierarchical branching networks,

    The law of Maximum Entropy Production

    Winiwarter's first law of genesis**
    ** The complexity of a selforganized system can only increase or remain constant.

    Winiwarter's second law of genesis***
    *** The rank frequency distribution of any selforganized system is of the PZM Pareto-Zipf-Mandelbrot type.

    Winiwarter & Cempel generalized "life symptoms" ****
    **** any selforganized system can be mapped on a hierarchy of birth and death processors formally equivalent to an ANN artificial neural network of the type multilayer perceptron (feedforward with backpropagation).

    Winiwarter Neural Networks ...

    Baillon-Moreno's unified Scientometric Model




Pareto-Zipf-Mandelbrot (parabolic fractal) distributions are observed in virtually all fields of research
covering the evolution from the big bang over the evolution of man
evolutionofman to the world wide web
.



empirical observations : self-organized* systems/  scalefree small-world networks and power laws



keywords  math:  long tail , "longtailed "/ "heavy tailed "/ "skewed" distributionsPareto law, Zipf's law, Zipf-Mandelbrot law, lognormal distribution , Yule-Simon distribution, Frechet Weibull distribution, rank-size rule,
parabolic fractal distribution
, **    
80/20 rulethe law of the vital few and the principle of factor sparsity ...
law of Gutenberg-Richter, Lotka's law, Bradford's law , Benford's law ... selforganized critically
power laws, scaling laws,   scalefree networks, "small world networks"  ...

all synonyms of the same statistical power law structure called PZM (Pareto-Zipf-Mandelbrot).


The common statistical feature of all the distribution types cited above are simple, they yield more of less straight lines in log log coordinates.
The mathematical forms of the distributions are more or less complicated. Statisticians have done extensive studies http://arxiv.org/PS_cache/arxiv/pdf/0706/0706.1062v1.pdf, trying to find out which distribution yields the best fit to a given data set. But they show, that if one distribution yields a good fit, then all the other distributions yield good fits also. (Error type three in the inquiry question, which is not what distribution is best, but why do we observe always similar distributions).
Let's apply Occam's razor and say that the most simple distribution will do it (the simple Pareto powerlaw, which is equivalent to Zipf's law or the rank size rule by simple inversion of coordinates. For discrete distributions (which are the case in most real examples) the Zipf-Mandelbrot or parabolic fractal distribution is the most simple form to prefer to complex constructs like Yule-Simon oir Frechet Weibull.
In the following we therefore speak of PZM (Pareto-Zipf-Mandelbrot or parabolic fractal distribution).




parabolicFractal1

** In the parabolic fractal distribution,  the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank. This can markedly improve the fit over a simple power-law relationship.

How to explain this to a non-mathematician?

Very simple:
Let's take the income size distribution of any country.  There are very few very rich billionaires, there are few rich millionaires, there are many middle class peope and the remaining vast majority are just plain poor. This inequality can be described by a mathematical distribution, which yields a straight line in log-log coordinates. (Pareto law).

Another example for PZM Pareto-Zipf-Mamdelbrot regularity. Let's take the city size distribution of a country and rank the citites in decreasing order of number of inhabitants. There are a few very big metropoles, there are a few big cities, there are many cities of medium size and the vast number of agglomerations are small towns. This inequality can be described by a mathematical distribution, which yields a straight line in log-log coordinates. (rank size rule).  etc... 

This asymmetric distribution is very different from the well known Gaussian bell shaped symmetric distribution. 

lognormal distribution
gaussPareto
normal Gaussian distribution versus Pareto-Zipf distribution
for those who persist to say they are the same mathematical structures, there is a major difference even in the second and third degree of a taylor development.
Pareto-Zipf-Mandelbrot (parabolic fractal) distributions  are scalefree.




In the parabolic fractal distribution the right tail of the poor is very much longer than the short left tail of the rich.  Therefore the term longtailed. For these distributions there is no such thing calculable like a mean or average income, since there is no symmetry and a value would be different for every arbitrary cutoff point in the ranking.

-
Note that there are only two families of mathematical distributions which do not change their form after the merger or split of system distributions :
1) the Gaussian  (Gauss folded with Gauss yields Gauss).
2) the Pareto-Zipf-Mandelbrot distribution  (Pareto folded with Pareto yields Pareto). see second law of genesis/
secondlawpfgenesis

For all the examples on this page we observe similar regularities of the Pareto-Zipf-Mandelbrot (parabolic fractal) type.

in the fields of:

big bang, astrophysics, geophysics, geology, geoscience, physics, metereology, phys-chem, biochemistry, biology, plant, animal, ecosystems, environment, social systems, transportation systems, economics, sociology, religion, linguistics, mechanical systems, communication systems, computer Tech, world wide web, scientometrics, brain,
neural networks ...


Abstract:

all networks revealing Pareto-Zipf-Mandelbrot (parabolic fractal) powerlaws are:

  1. energy/information transformation systems
    -
  2. built of basic energy/information transformation processors that are born and run to  death in an irreversible way (birth and death processors)
    energproc
    a part of the input energy rate is irreversibly accumulated in an internal reservoir from birth to death of the processor.


    FracNaturEnProc1

    fractal hierarchy of birth and death processors in nature





    Physics, complexity and causality

    With this view, the higher levels in the
    hierarchy of complexity have autonomous
    causal powers that are functionally independent
    of lower-level processes. Topdown
    causation takes place as well as
    bottom-up action, with higher-level contexts
    determining the outcome of lowerlevel
    functioning, and even modifying the
    nature of lower-level constituents.

    George F. R. Ellis
    NATURE|Vol 435|9 June 2005
  3. all processors are linked in a complex "small world"  network mappable on a multilayer perceptron network of artificial neurons*

    smallworldnetwork 

    Small worlds, between perfect order and chaos;
    the first graph is completely ordered, the graph in the middle is a "small world" graph, the graph at the right is complete random.





     
    smallworldperceptron0  
    smallworldperceptron1                                                                     

    Note: when a small world network is mapped on a specific multilayer feedforward network (multilayer perceptron) the neurons of the top layer constitute virtual nodes of the binary "shortcut" links of the small world.

  4. it's the global field generated by all processors that "drives" the process of evolution based on energy optimization specific to the level of evolution :
    GUT, gravitation, strong nuclear, weak nuclear, electo-magnetic, chemical, geothermal, wind, water, fire, genetic code, words, written codes, computer codes ...
    -
  5. ritualization : repetitive use of pathways, Hebb's rule  ( cells that fire together wire together) and the Pareto frontier (You can't make many people better of without making many people worse of)
     hardwire the networks information flow into "engrams"**, like timetables hardwire a railroad, or air transportation network.
hebbsRule Hebb's rule  "cells that fire together, wire together"



*Hebbian theory has been the primary basis for the conventional view that when analyzed from a holistic level, engrams are neuronal nets or neural networks. Additional ideas regarding cell assembly theory and its role in forming engrams, along the lines of the concept of auto-association, described as follows:
**  "If the inputs to a system cause the same pattern of activity to occur repeatedly, the set of active elements constituting that pattern will become increasingly strongly interassociated. That is, each element will tend to turn on every other element and (with negative weights) to turn off the elements that do not form part of the pattern. To put it another way, the pattern as a whole will become 'auto-associated'. We may call a learned (auto-associated) pattern an engram." (Op cit, p44;) Gordon Allport
trees1
trees2

All the tree networks above show empirically observable Pareto-Zipf-Mandelbrot (parabolic fractal) statistical structure.

Autocatalytic Hypercycles usually follow tree engrams in the process of evolution.

hypercycles

"Life and mind have a common abstract pattern or set of basic organizational properties. The functional properties characteristic of mind are an enriched version of the functional properties that are fundamental to life in general. Mind is literally life-like. "

                   
                                             Godfrey-Smith, P. (1996). Complexity and the Function of Mind in Nature. Cambridge: Cambridge University Press.

"The Universe and Life are literally mind-like. "

                   
                                             Peter Winiwarter (2008). Network Nature. www.bordalierinstitute.com




1) empirical observations : self-organized* systems/  scalefree small-world networks and power laws

 

*The term "self-organizing" seems to have been first introduced in 1947 by the psychiatrist and engineer W. Ross Ashby. It was taken up by the cyberneticians Heinz von Foerster, Gordon Pask, Stafford Beer and Norbert Wiener himself in the second edition of his "Cybernetics: or Control and Communication in the Animal and the Machine" (MIT Press 1961). Self-organization as a word and concept was used by those associated with general systems theory in the 1960s, but did not become commonplace in the scientific literature until its adoption by physicists and researchers in the field of complex systems in the 1970s and 1980s.


Pareto-Zipf-Mandelbrot (parabolic fractal) distributions are observed in virtually all fields of research
covering the evolution from the big bang over the evolution of man
evolutionofman to the world wide web
.




Short before the big bang

bibBangbefore the big bang 2hands







"The intelligence lies in your hands."
  elsa genèse, sculpteur    


                yourcenarMains        yourcenarTete

                     "Marguerite Yourcenar" (details)  monumental sculpture of elsa genese





big bang eras
On each level of evolution there is the emergence of a new type of energy field which creates its proper engrams assembled from existing building blocks of the lower levels prior in evolution.



in the fields of:





- astrophysics :




star evolutioncarboncycle1{Eigen & Schuster 1979:3}



chart of nuclides
chemical elements

Note the extreme abundance of Hydrogen followed by Helium and the extreme rare abundance of heavy neutron-rich elements like Uranium (a difference of 13 orders of magnitude) wich are produced during successive generations of nucleosynthesis in massive stars.


 nuclear abundance size distribution from Hydrogen to Uranium (nucleosynthesis in stars)

PZM (Pareto-Zipf-Mandelbrot, parabolic fractal) nuclear abundance distributions are observed for all stars comprising the sun and the universe as a hole.


starsize2starEvolution

star size distributions are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

There are few giant stars, many medium size stars like the sun and a full bunch of small dwarfs.







galaxy

galaxy

galaxiesCollection

galaxy size distributions
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

butterflies

A zebra's stripes, a seashell's spirals, a butterfly's wings: these are all examples of patterns in nature. The formation of patterns is a puzzle for mathematicians and biologists alike. How does the delicate design of a butterfly's wings come from a single fertilized egg? How does pattern emerge out of no pattern?  http://www.sciencedaily.com/releases/2008/06/080619111748.htm



There are a few giant galaxies, many medium size galaxies and a full bunch of small galaxies. The pictures above are not scaled to actual size.


galaxy cluster

galaxy cluster size distribution are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

Rare Monsterclusters, few big clusters, an awfull lot of small clusters




 planet

planet scale

 planet size, satellite of planet (moon) size distributions are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

Big planets are rare, there are some medium size planets, while small planets and moons are frequent.


meteoritemeteoriteSwarm

 meteorite size distributions
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)


moon crater
                           
 moon crater size distribution 
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

Since meorite size distributions are fo the PZM type, moon crater size distributions are of the Pareto-Zipf-Mandlebrot type.



solar flares


the brain of the sun, peak gamma ray intensity distribution of solar flares 
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

solarburst

There are a few very big bursts, several medium size bursts and the vast majority of small bursts.




sun_evoluiton

sun evolution (an example of a birth & death processor life cycle)




- geophysics, geology :


earthquaketectonicMovements

tectonic movements and earthquake energy size distribution are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)
(law of Gutenberg-Richter)  
http://scimaps.org/dev/map_detail.php?map_id=160

Remember the "big one" to come, large earthquakes are rare, medium size quakes are more frequent and small quakes are almost anytime and everywhere.

PZM (Pareto-Zipf-Mandelbrot, parabolic fractal) earthquake energy size distributions are observed for all regions of the globe .





geothermal energy

 vulcanic eruption size distribution are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

The big ones leave traces in the history of earth and mankind (Stromboli).





        
 island sizeIslands Indonesia

 island size distribution are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

There is the biggest island of Crete, next size there is Rhodes, there a many medium size islands and a vast number of small islands. (why?) idem Indonesian islands





riverbasin2riversize3
riverdelta2

river size distribution are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

There are few large streams like the Missisippi, there are many rivers and millions of ways for water to run to the sea. (the final attractor of the network)




floods2
floodsize

flood size distribution
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

Floods are like earthquakes, humanity remembers only the big ones. 





lakes2lake size
lake size distribution are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

Big lakes are rare, while small lakes and ponds are everywhere. (the neural networks of water)





geoscience :


sedimentssediments2

cosmic and terrestrial dust seston sediment size distribution are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)









oilreserves1parabolicFractal

field size distribution of oil reserves
 are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

We would prefer an equal distribution among countries, but network Nature goes it's own itelligent way.






- physics :



 
sand avalanchehourglass

size distribution of sand particle avalanches are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

(This gave rise to a topic burst on the topic selforganized critically of granular media where the phycisit Pierre BAK tried to explain "How Nature works") ...
Natural networks are granular, but they don't simply function like piles of sand.





percollationNetwork

percollation network

you will find this phenomenon on any level of evolution.


meteorology



hurricane

hurricane energy size distribution are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

Prigogine could have taken this example for "self-organization" instead of exotic chemical reactions like the
Belousov-Zhabotinsky reaction.






weapon systems :


nuclearExplosion

nuclear arm explosion energy size distributions
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

"mankind and technology are part of nature and the laws of nature apply to mankind and technology."  Peter Winiwarter.

hurricans and nuclear arms energy
The fatality distribution of tornadoes (/), floods (2), hurricanes (3), and earthquakes (4) in the 20th century United States.

tornadoes, floods, hurricans, earthquake & nuclear arms energy size distributions are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

Each network has it's own characteristic exponent.

"There are laws of destruction, but they differ in kind and methods"  Peter Winiwarter








phys-chem  :



hypercycleevolutionhypercycleArt

hypercycle chemical element frequency distributions are of the PZM Pareto-Zipf-Mandelbrot type (hyperbolic fractal)  {Eigen & Schuster 1979:86}

"the schematic view of the evolution of a hypercycle can be considered as the prototype for every evolution and life."  Peter Winiwarter








bio-chemistry :




chemical comppunds     proteinNetwork
E.coli metabolic network

The E.coli metabolic network is scalefree ( PZM Pareto-Zipf-Mandelbrot type, parabolic fractal) and has small-world properties

yeast protein interaction network
The yeast protein interaction network has a scalefree topology (Pareto-Zipf Mandelbrot distribution)
The scale-free nature of protein interaction networks is a generic feature of all organisms.







- biology 



evolutionTreeDarwinTreeofLifeDarwin Tree of Life


population size distribution of species, species size distribution of genus, genera size distribution of biological family are of the PZM Pareto-Zipf-Mandelbrot type (hyperbolic fractal)
(Yules law) ...

PZM (Pareto-Zipf-Mandelbrot, parabolic fractal) distributions observed for all species at all times of biological evolution


"In terms of genetic evolution mankind is close to big apes, in terms of social evolution mankind is much closer to ants , termites and bees."  Peter Winiwarter
evolutioanryTree

evolutionary leaps :


appearance of life, eucariotes, pluricellular, tetrapodic, homeothermic, vivipartide, primates
rank-time distribution
evolutioanryLeaps 

animal :




 animalEvolution


PZM (Pareto-Zipf-Mandelbrot, parabolic fractal) distributions observed for all species at all times of biological evolution


"its the same underlying computational algorithms which drive evolution. Mutations are not random, they are computed."  Peter Winiwarter


plant :

                       
   
tree branches

   branch size distribution of trees are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

"I see trees everywhere"  Peter Winiwarter


treestems


tree stem diameter distributions 
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

"most citations are truncated."  Peter Winiwarter



leaveSizeDistribution


leaf area size distribution  
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)


"        'leaf me alone' is the most fundamental principle of plants."   Peter Winiwarter


metabolicrateAnimals2
the same scaling law is observed over 27 orders of magnitude from microbacteria to whales.

metabolism3bloodvessel  bloodvessel





aortahierarchy


aorta blood vessel hierarchy, see multylayer perceptron below




WestMinmaxPriciple

metabolic pathways


metabolic rate in animals of all size (allometric scaling distribution)

  1. MINIMIZE ENERGY DISSIPATED IN THE NETWORKS
  2. MAXIMISE THE SCALING OF THEIR AREA OF INTERFACE WITH THEIR RESOURCE ENVIRONMENT
That's the INTELLIGENCE of Nature.



heartRate

heart rate of all animals follow a power law distribution

"The number of your heartbeats until your death are counted by your weight."   Peter Winiwarter



bloodvascularsystem

blood vascular system
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

"Women have particularly vascular networks in their lips."   Peter Winiwarter

note:  I like this citation





- ecosystems  environment :


foodweb1foodweb2

 

biomass-size distribution of aquatic ecosystems (trophic web or foodweb)
Winiwarter and Vidondo modelled the ecosystem evolution of the lake Constance by a neural network of the feedforward type with backpropagation (multilayer perceptron).

PZM (Pareto-Zipf-Mandelbrot, parabolic fractal) biomass-size distributions are observed in all oceans and all lakes






forest fire

distribution of areas burnt in forest fires are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

Remember the mooncrater size distribution caused by meteorites, forest fires scale like PZM.


 
http://www.bordalierinstitute.com/target1.html#religionvegetationpatches

patches of vegetation  
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)







- social systems, transportation systems :

"Marguerite Youcenar" monumental sculpture of elsa genese
ese city size





population density     
cities2006          citysizeUS
largest US cities in ordinary scale (straight line in log-log scale)

human settlement size distribution, city size distribution (rank-size rule)

The numbers speak for themselves.

Dans les sciences contemporaines, l’intérêt porté à la complexité provient de l’insuffisance de la démarche analytique classique à expliquer le réel. La complexité postule qu’il est impossible de décrire, sans réductionisme, certains phénomènes indécomposables et que le Tout n’est pas réductible à des unités élémentaires.
La notion de complexité est utilisée à deux niveaux différents : la complexité dans la constitution des choses : de la cellule à l’organisme et la complexité dans l’interprétation qui fait référence aux relations entre les composantes d’un système.
En Géographie existe une tension entre la nécessité pour produire des connaissances nouvelles de mettre de l’ordre dans les phénomènes, en risquant de fabriquer du simplifié et, de l’autre la dilution des schémas explicatifs en les intégrant dans des totalités imprécises. La démarche analytique disjonctive est fréquente en Géographie du fait de postures méthodologiques qui tendent à privilégier le plan à tiroirs, la classification et la fragmentation territoriale par rapport à l’explication et, à confondre délimitation spatiale de l’objet d’étude et définition dudit objet.
L’adoption du paradigme de la complexité implique une remise en cause de la causalité linéaire utilisée dans les schémas explicatifs traditionnels. A un enchaînement procédant d’une cause première, et s’appuyant à chaque maillon du raisonnement sur une monocausalité, on substitue à la fois une multicausalité et une démarche en boucle où l’effet rétroagit sur la cause. Dans un raisonnement en boucle, chaque composante est un maillon indispensable, un moment dans la production de la connaissance qui se génère elle-même, en même temps qu’elle génère l’explication générale.
La pensée de la complexité nécessite d’accorder autant d’importance à l’agencement des objets entre eux, aux interactions et aux types de liaisons entre ces objets qu’aux objets eux-mêmes. Elle suppose que dans un système existe une autonomie organisationnelle qui permette une reproduction et une transformation de ses constituants, c’est à dire une auto-organisation. Le principe de complexité permet d’interpréter en Géographie le fonctionnement des systèmes dynamiques tels que les villes, les régions ou les géosytèmes. L’intelligibilité du complexe se fait par la modélisation.
Dès lors, l’objectif des sciences sociales ne consiste plus à réduire le complexe au simple, mais à traduire le complexe en théorie.
Dans les sciences contemporaines, l’intérêt porté à la complexité provient de l’insuffisance de la démarche analytique classique à expliquer le réel. La complexité postule qu’il est impossible de décrire, sans réductionisme, certains phénomènes indécomposables et que le Tout n’est pas réductible à des unités élémentaires.
La notion de complexité est utilisée à deux niveaux différents : la complexité dans la constitution des choses : de la cellule à l’organisme et la complexité dans l’interprétation qui fait référence aux relations entre les composantes d’un système.

Dès lors, l’objectif des sciences sociales ne consiste plus à réduire le complexe au simple, mais à traduire le complexe en théorie.

"The objective of social sciences does not consist any more in the reduction of complex to simple,  but in the translation of complex into theory."
 Peter Winiwarter




PZM (Pareto-Zipf-Mandelbrot, parabolic fractal) city-size distributions are observed for all countries of the world at all times

"In terms of genetic evolution mankind is close to big apes, in terms of social evolution mankind is much closer to ants , termites and bees."  Peter Winiwarter

transportation Los Angeles"Marguerite Youcenar" monumental sculpture of elsa geneseduchamp"Marguerite Youcenar" monumental sculpture of elsa genese
 public transportation networks ... (Marcel Duchamp  "Network stoppages")

PZM (Pareto-Zipf-Mandelbrot, parabolic fractal) Power laws are observed for the world's 12 major metropolitan public transportation network systems



railroadnetworks

railroad networks
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)




airportnetwork
"Marguerite Youcenar" monumental sculpture of elsa geneseairportnet2

airport2 The world-wide airport network is a
small-world
network for which the number of direct connections
k to a given city (degree) has a cumulative distribution
P(> k) that decays as a truncated power-law
P(> k) ∝ k−α f(k/k×) , (1)

where α = 1.0 ± 0.1 is the power-law exponent, f(u) is
a truncation function, and k× is a crossover value that
depends on the size S of the network as k× ∼ S0.4.




powergrid

power grid distribution lines
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

There are very few very high voltage power lines, there are many high voltage lines and a billion of low voltage (220 or 110 V ) lines to the final consumers.
The network is a typical example of a small world hierarchical branching system optimized by man control.








economics , income systems :


"Marguerite Youcenar" monumental sculpture of elsa genese

fortuneyacht Bill Gates
yacht Paul Allenyacht Larry Ellison
boatpeopleyachtWiniwarter1


       
      
the world's billionaires, follow a PZM Pareto-Zipf-Mandelbrot (parabolic fractal)  distribution from the "toyboys": yachts of Bill Gates,  Paul Allen and Larry Ellison  to boatpeople ... and solitary outsiders like Peter Winiwarter

fortunebillionaires2fortuneBillioinaires            

 individual income size distribution (Pareto law) 

PZM (Pareto-Zipf-Mandelbrot, parabolic fractal) income size distributions are observed for all countries at all times independant of the political system from the democratic U.S. to feudal renaissance princedoms
(Karl Marx ignored this law of nature).


pareto fronteer:
You can't make many people better of without making many people worse of.





firm size fortune 500

firm size distributions 
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)
(Fortune 500 asset size distribution of the lagest U.S. industrial corporations 1965, 1975, 1981)
note the persistance of the slope despite growth, birth and death of corporations during the period of observation.

                        

capitalization

distributions of capitalization
, distribution of number of employes ...
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

 any country, any political system,  Fortune 500 (1000) database,







- religion 


personalnetwork hologram universe2
social network of person or goddess



jesusNetwork

Jesus network shows a
PZM Pareto-Zipf-Mandelbrot type (parabolic fractal) degree distribution


world!relegions size distribution of religous groups are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)







   greekMythologygreekgoodsMonsters


greek & roman mythology individuals (gods and heroes) network 
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)




sociology :




antswarm


antswarm
antforagingtrails


ant foraging trails are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)




sexualdeseasepropagation

sexual desease propagation






peter, lotte, fritzl

 genealogical trees are of the PZM Pareto-Zipf-Mandelbrot type (hyperbolic fractal)

my uncle Peter (died in world war II in Russia), my ant Lotte and my father "Fritzl" von WINIWARTER to whom I dedicate this work


  
genealogicalTree

 family name frequency distributions are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

extremely frequent names SMITH, BROWN,  extremely rare names WINIWARTER






moviecoactor

movie co-actor networks on the Internet Movie Data base are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

In modern mathematics, the center of the network of coauthorship is considered to be P. Erdős, resulting in the so-called Erdős number. In movies, Kevin Bacon is often mentioned as the center of the movie universe, but a recent study (Reynolds) has shown Christopher Lee to be the actual center. Both actors have co-starred with Julius LeFlore, so the Lee-Bacon distance is two.




warbattledeathswarcimeteries

battle deaths per war distributions
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)






- linguistics :



language Babellanguage network

word frequency distributions  (Zipf's law)

PZM (Pareto-Zipf-Mandelbrot, parabolic fractal) distributions are observed for the word frequencies of all texts of all languages, all times, any age of author, even the bubbling of babies show the same Pareto-Zipf-Mandelbrot distribution with a slope of 1.00  ...







- mechanical systems :



machine

 distribution of energy dissipation symptoms in machine networks,  railroad networks ... are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)






- communication systems :




telecommunication

distribution of telephone calls received
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)







bestseller

distribution of numbers of bestselling books
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)




- computer tech :





fileSizeC


file size distribution on hard disk drive (my computer drive C), network size, length distribution in jobs assigned to supercomputers ... are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)

Three different types of visualization of the same underlying directory tree structure.
fileDirectoryTree1
fileDirectoryTree2fileDirectoryTree3


evolutionLanguages

the network of computer languages
...
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal) (hypothesis)computerEvolution

computer evolution (Apple) written on a Mac Mini runnung under Windows Vista Pro


software development

codeSwarm


A moment from the Eclipse project.
"I've been studying software projects for a while now. Not the programming, but the people -- the way they interact with each other through collaboration and communication. My investigations have always been visual: I've built applications that create pictures of what is happening within software projects. But they have always had a rigid structure to them. Organic information visualization, coined by Ben Fry, is a different approach to information visualization. It eschews traditional data confinement in space and lets the elements play together in freeform and unpredictable ways.
This visualization, called code_swarm, shows the history of commits in a software project. A commit happens when a developer makes changes to the code or documents and transfers them into the central project repository. Both developers and files are represented as moving elements. When a developer commits a file, it lights up and flies towards that developer. Files are colored according to their purpose, such as whether they are source code or a document. If files or developers have not been active for a while, they will fade away."


                         

- world wide web:

`networkevolution 


user visit to site distribution, site-size ranks,
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)
                                                                                                                   
local viewwebgraph
hyperbolic view
worldwidewebstructure
webgraphwebSplatMap
 
webgraph in/out links,

File size distribution of Internet traffic,
frequency of numbers occuring in the web,
distribution of web advertizing budgets ...
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)






scientometrics,

  research and publications :




researchNodes


research nodes , a hyperbolic fractal


note the hypercycle structure of the network


the scientometric model

Summary  
A unified scientometric model has been developed on the basis of seven principles: the actor-network principle, the translation principle, the spatial principle, the quantativity principle, the composition principle, the centre-periphery or nucleation principle, and the unified principle of cumulative advantages.
The paradigm of the fractal model has been expanded by introducing the concept of fractality index and transfractality.
In this work, as the first demonstration of the power of the model proposed, all the bibliometric laws known and all their mathematical expressions are deduced, both the structural distributions (Zipf, Bradford and Lotka) as well as the Price's Law of the exponential growth of science and Brookes' and Avramescu's Laws of ageing.


http://www.virtualknowledgestudio.nl/staff/andrea-scharnhorst/documents/heraeus_silverberg.pdf



science1

astrophysics astrophysics astrophysics phycics geophysics geology geoscience meteorology phys-chem biochemistry biology plant animal environment ecosystems social systems transportation systems economics sociology linguistics communication systems computer tech world wide web brain neural networks biochemistry math math statistical distributions education topic map of science with links (click on the subject) to PZM Pareto-Zipf-Mandelbrot (parabolic fractal) distributions.

In virtually all fields of science with the exception of Psychology and Psychiatry, desease & treatment, cancer, virology and Infectious Diseases* , Law and Policy
we observe "generalized life symptoms" of the PZM family.

Note that in linguistics the only known exceptions to Zipf"s law (which holds for all languages at all times at any age) are the writings of schizophrenics and scientific literature.

* new works are emerging

Journal of Theoretical Biology : Scaling properties of childhood ...

Therefore, SEIR simulates infectious diseases at the collective level by ..... the relative frequency in the use of words in language (Zipf's law) or the ..



co-authorshipNetwork






co-authorship networks are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)  see animated graph  http://iv.slis.indiana.edu/ref/iv04contest/Ke-Borner-Viswanath.gif



scientometrics

 citation frequency distributions (Lotka's law) are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal) journalometry (Bradford's law),



co-citation
 co-citation graph of 394 articles
"the DNA/RNA and proteins of research papers"






topicBursts


 publication avalanche size distribution (topic bursts) are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)



education :



researchInstitutions


education: distribution of research institutions are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)







innovation : 

innovation1

innovation size distribution
are of the PZM Pareto-Zipf-Mandelbrot type (parabolic fractal)















- Brain, neural networks :    neuron2
brainUniverse



cerebellum
brainEvolution

Short-term memory uses small world networks between neurons to remember this sentence.



neural networks  neuron 


UniverseNeuralNetwork


the universe: a self-similar hierarchy of intelligent neural networks



 multilayer perceptron error distributions in cyclic learning process of feedforward and backpropagation  follow a PZM Pareto-Zipf-Mandelbrot (parabolic fractal) distribution as a single unit and co-operating in a network.



etc. etc. ...

history of science

Conceptualizing science

http://scimaps.org/dev/map_detail.php?map_id=163


metaphysics :



cosmicDomains
cosmic domains, a metaphysical hierachy after R.A.Monroe and C.Cempel, beyond fractal reality

We share the world view of Heinz von Förster, the world (what we call "reality") is a construction of our mind. (constrained by the form of our conceptual containers, see Semiotica article on the periodic system of system concepts).
Science is the collective construction of world views based on majority consensus.
What's beyond is called metaphysics and as Kant said, "metaphysics is like a dark ocean without shore and without light-tower." (better not talk about it)
We cannot say anything about it except believe systems.
However Kant also said, "if you drive metaphysics out of the front door, it will enter through the back window."
That's why we started and ended this illustrated scientific walk through evolution from the big bang to the world wide web by a metaphysical slide.







2) theoretical approaches

nebouleuse theory
"We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that is not crazy enough. "

Niels Bohr on Pauli's theory of elementary particles
from Arne A. Wyller's The Planetary Mind

Niels Bohr on Pauli's theory of elementary particles
from Arne A. Wyller's The Planetary Mind
Niels Bohr on Pauli's theory of elementary particles
from Arne A. Wyller's The Planetary Mind.





The key question I ask myself since 25 years of research :

"Why do all the above complex systems / networks show identical statistical structure (Pareto-Zipf-Mandelbrot distribution), which is empirically observable?"

keywords:

Zipf's principle of least effort
,

Pareto efficiency Pareto frontier
,

Holland Gell-Mann complex adaptive systems

Odum's maximum power principle*
*Chen (2006) has located the origin of the statement of maximum power as a formal principle in a tentative proposal by Alfred J. Lotka (1922a, b). Lotka's statement sought to explain the Darwinian notion of evolution with reference to a physical principle. Lotka's work was subsequently developed by the systems ecologist Howard T. Odum in collaboration with the Chemical Engineer Richard C. Pinkerton, and later advanced by the Engineer Myron Tribus.
Gilliland noted that these difficulties in analysis in turn required some new theory to adequately explain the interactions and transactions of these different energies (different concentrations of fuels, labour and environmental forces). Gilliland (Gilliland 1978, p. 101) suggested that Odum's statement of the maximum power principle (H.T.Odum 1978, pp. 54-87) was, perhaps, an adequate expression of the requisite theory:
That theory, as it is expressed by the maximum power principle, addresses the empirical question of why systems of any type or size organize themselves into the patterns observed. Such a question assumes that physical laws govern system function. It does not assume, for example, that the system comprising economic production is driven by consumers; rather that the whole cycle of production-consumption is structured and driven by physical laws.

West's min-max principle for hierarchical branching networks,

Winiwarter's first law of genesis**
** The complexity of a selforganized system can only increase or remain constant.

Winiwarter's second law of genesis***
*** The rank frequency distribution of any selforganized system is of the PZM Pareto-Zipf-Mandelbrot type.

Winiwarter & Cempel generalized "life symptoms" ****
**** any selforganized system can be mapped on a hierarchy of birth and death processors formally equivalent to an ANN artificial neural network of the type multilayer perceptron (feedforward with backpropagation).

Winiwarter Neural Networks ...

Baillon-Moreno's unified Scientometric Model

Leydesdorff and Hellsten    MeasuringMeaning : Monarch butterflies, Frankenfoods, and stem-cell therapies
Published in Scientometrics 67(2), 2006, 231-258. Abstract: Co-words have been considered as carriers of meaning across different domains in studies of science, technology, and society. Words and co-words, however, obtain meaning in sentences, and sentences obtain meaning in their contexts of use. At the science/society interface, words can be expected to have different meanings: the codes of communication that provide meaning to words differ on the varying sides of the interface. Furthermore, meanings and interfaces may change over time. Given this structuring of meaning across interfaces and over time, we distinguish between metaphors and diaphors as reflexive mechanisms that facilitate the translation between contexts. Our empirical focus is on three recent scientific controversies: Monarch butterflies, Frankenfoods, and stem-cell therapies. This study explores new avenues that relate the study of co-word analysis in context with the sociological quest for the analysis and processing of meaning.

Baillon-Moreno's  Structure and dynamics of scientific networks:

Abstract Here, the quantitative theory of translation is shown to be of great utility in describing scientific networks. In fact, we deduce a new Zipf's Law for the descriptors of a set of documents, based on the concepts of centres of interest and of irreversible parallel translations. This new law can be generalized to other phenomena, such as the distribution of the sizes of cocitation clusters. Finally, we have established the model, for descriptor presence in a network, which closely fits the values recorded.





Abstract:
For income distributions, city-size distributions and word frequencies we observe long-tailed distribution power laws of the PARETO-ZIPF-MANDELBROT (parabolic fractal) type.

In fact on all levels of evolution from the big bang to the world wide web - for properties of units belonging to the same astrophysical, geological, biological, ecological, urban, social, linguistic, economic, mechanical machinery or computer system - we observe similar regularities referred to as Pareto law, Zipfs law or rank-size law.
We give a short historical overview over the discovery of these empirical regularities. Then we review a variety of theoretical tentatives, none of which can explain that similar regularities are observed in such an incredible wide range of scientific research areas.
Finally, departing from a very specific model - describing tribo-vibro-acoustic processes in machines - we propose a generalized theoretical framework in terms of energy transformation with limited internal energy dissipation capacity, which is applicable to all fields.
The proposed model unifies a large variety of concepts and applies a coherent terminology to fields, which have at first sight nothing in common.
For the observed generalized life symptoms, theoretical predictions can be compared with past and future empirical observations.
What is most important is the model's inference power: from the observations of a set of units at a given moment of life-time (a snapshot of the system), one can predict the average behaviour of a single unit over its entire life-time.

Following the model of "generalized life symptoms" of complex systems / networks of "birth and death processors" we propose a unifying paradigm, identifying all observed systems reveiling PZM (Pareto-Zipf-Mandelbrot) distributions as internal statistical structure:

all networks revealing Pareto-Zipf-Mandelbrot powerlaws are:

  1. energy/information transformation systems
    -
  2. built of basic energy/information transformation processors that are born and run to  death in an irreversible way (birth and death processors)
    birthanddeathprocessorgrowthcurvebirthanddeathstar birth and death of star

  1. all processors are linked in a complex "small world" network mappable on a multilayer perceptron network of artificial neurons
    smallworldnetwork
  2. it's the global field generated by all processors that "drives" the process of evolution based on energy optimization specific to the level of evolution :
    GUT, gravitation, strong nuclear, weak nuclear, electo-magnetic, chemical, geothermal, wind, water, fire, genetic code, words, written codes, computer codes ...
    -
  3. ritualization (repetitive use of pathways, Hebb's rule http://en.wikipedia.org/wiki/Hebb%27s_rule) hardwires the networks information flow, like timetables hardwire a railroad, or air transportation network.
hebbsRule Hebb's rule  "cells that fire together, wire together"

Hebbian theory has been the primary basis for the conventional view that when analyzed from a holistic level, engrams are neuronal nets or neural networks.

Gordon Allport posits additional ideas regarding cell assembly theory and its role in forming engrams, along the lines of the concept of auto-association, described as follows:

"If the inputs to a system cause the same pattern of activity to occur repeatedly, the set of active elements constituting that pattern will become increasingly strongly interassociated. That is, each element will tend to turn on every other element and (with negative weights) to turn off the elements that do not form part of the pattern. To put it another way, the pattern as a whole will become 'auto-associated'. We may call a learned (auto-associated) pattern an engram." (Op cit, p44;)


Any  selforganized network ("small world") can be mapped on a network of artificial neurons of the multilayer perceptron type. These networks recycle information in a learning process based on feed forward and backpropagation cycles.


treelife



fractalNetworks


Hence the properties of this type of artificial neural networks are common to all systems / networks for which we observe PZM (Pareto-Zipf-Mandelbrot, parabolic fractal) distributions:
  1. selforganized systems / small world  networks  / the universe have memory  
    (engrams :  tree structure, hypercycle structure)
  2. selforganized systems / small world networks / the universe  are learning  
    (Hebb's rule, Pareto frontier)
  3. selforganized systems / small world networks / the universe are intelligent
    (global energy/information optimization)






Links to publications by the author of this page Peter WINIWARTER :

citationWebs  (work in progress)


    References 

1.    Clauset, A., Shalizi, C. R., & Newman, M. (2007, 7 Jun). Power-law distributions in empirical data. physics.data-an. Retrieved from arxiv.org: http://arxiv.org/PS_cache/arxiv/pdf/0706/0706.1062v1.pdf
        example for articles on statistical analysis of power laws

2.    Guimera, R., & Amaral L.A.N. (2004). Modeling the world-wide airport network [Airport network]. The European Physical Journal B, 38, 381-385. http://www.bordalierinstitute.com/airport.pdf
         example for articles on analysis of "smal world" networks

3.    Eigen, M., & Schuster, P. (1979). The Hypercycle: A Principle of Natural Self-Organization (p. 92). Berlin, Heidelberg, New York: Springer-Verlag.
          example for a remarkable book used in this research  http://en.wikipedia.org/wiki/Manfred_Eigen


4.    Roehner, B., & Winiwarter, P. (1985). AGGREGATION OF INDEPENDENT PARETIAN RANDOM VARIABLES [probability]. Advances in Applied Probabilities, 17, 465-469. http://www.bordalierinstitute.com/probability.pdf
          statistical stability of Pareto distributions

5.    Winiwarter, P. (1983/108-112). THE GENESIS MODEL - PART II: FREQUENCY DISTRIBUTIONS OF ELEMENTS IN SELFORGANIZED SYSTEMS [GenesisModel 2]. Speculations in Science and Technology, 6(2). 103-112. http://www.bordalierinstitute.com/genesisModel2.pdf

6.    Winiwarter, P. (1983). THE GENESIS MODEL PART I: COMPLEXITY, A MEASURE FOR THE EVOLUTION OF SELF-ORGANISED SYSTEMS OF MATTER [GenesisModel 1]. Speculations in Science and Technology, 6(1), 11-20. http://www.bordalierinstitute.com/genesisModel1.pdf
         a quantitative measure of complexity, complexity can only increase or remain constant.

7.    Winiwarter, P. (1984). HOMEOSTATIC TENDENCIES OF FIRM-SIZE DISTRIBUTIONS AND THE EVOLUTION ECONOMIC SYSTEMS [firmsize]. In  AFCET (Ed.), Cybernetics and Systems (pp. 999-1005). 6th International Congress of Systems WOGSadfasdf, 2. http://www.bordalierinstitute.com/firmsize.pdf
        
8.    Winiwarter, P. (1986). AUTOGNOSIS: THE THEORY OF HIERARCHICAL SELF-IMAGE BUILDING SYSTEMS [Autognosis]. In  Society for General Systems Research (Ed.), Mental Images, Values and Reality (Vol. A, pp. A21-A31). 30TH ANNUAL MEETING. http://www.bordalierinstitute.com/autognosis.pdf

9.    Winiwarter, P. (1986). CONCEPTS OF SELF-ORGANIZATION () SELFORGANIZATION OF CONCEPTS [Selforg]. In  Society for General Systems Research (Ed.), Mental Images, Values and Reality (Vol. D, pp. D62-D76). 30TH ANNUAL MEETING. http://www.bordalierinstitute.com/self-organization.pdf

10.    Winiwarter, P. (1986). NATURAL NUCLEAR COMPUTERS: A MODEL FOR HUMAN INTELLIGENCE [Nuclear computers]. In  Society for General Systems Research (Ed.), Mental Images, Values and Reality (Vol. D, pp. D1-D9). 30TH ANNUAL MEETING. http://www.bordalierinstitute.com/starcomputations.pdf

11.    Winiwarter, P. (1999). A periodic System of System concepts [Conceptual framework]. Semiotica, 125(1/3), 47-62. http://www.bordalierinstitute.com/semiotica.pdf

12.    Winiwarter, P., & Cempel, C. (1992). Life Symptoms: The Behaviour of Open Systems with limited Dissipation Capacity [Life symptoms]. Systems Research, 9(4), 9-34. http://www.bordalierinstitute.com/lifesymptoms.pdf
          energy transformation and dissipation capacity

13.    Winiwarter, P., & Cempel, C. From Trophic Webs and Hierarchical Energy Transformation Systems to Multilayer Neural Networks [GenesisModel 3]. (1996). . In PPSN IV. Berlin: http://www.bordalierinstitute.com/neuralnetworks.pdf
           equivalence of a network of binary threashold "birth and death processors" and a "multilayer perceptron" artificial neural network

14.    Winiwarter,P., & Cempel, C. Evolutionary Hiearchies and Energy Processing in Nature [evolutioniary hierachies].(1996)In Austrian Society for Cybernetic Studies. Trappl, R. (Ed.), Vol.2, p 26-31.  http://www.bordalierinstitute.com/evolutionaryhierarchies.pdf
           coevolution of macroscopic and microscopic hierarchies


Some useful links of thousands:  WEB links

First back to homepage | back | next | Last
bordalier institute