Next: Multi-Agent Modeling of Reaction-Diffusion Systems
Up: Table of contents
Previous: Logistic Systems

One of the early hints of complexity was the observation in 1897 by Pareto that the wealth of individuals spreads over many orders of magnitude (as opposed to the size of a person which ranges roughly between 1 meter and 2 meters). The dynamics of the social wealth is then not dominated by the typical individual but by a small class of very rich people. Mathematically one realized that instead of the usual fixed scale distributions (Gaussian, exponential), the wealth follows a 'power law' distribution. Moreover, in spite of the wide fluctuations in the average wealth during crises, booms, revolutions, the exponent of the power laws remained between narrow bounds for the last 100 years.

Similar effects [7] were observed in a very wide range of measurements: meteorite sizes, earthquakes, word frequences and lately internet links. In all these systems, the presence of power laws constitutes a conceptual bridge between the microscopic elementary interactions and the macroscopic emergent properties.

Recently, attention has turned to the internet which seems to display quite a number of power-law distributions: the number of visits to a site [4], the number of pages within a site [5], and the number of links to a page [6], to name a few..

We will see in detail in this report that the autocatalytic character of the microscopic interactions governing these systems can explain this behavior in a generic unified way.

A quick plausibility argument is based on the observation that a dynamics in which the changes in the elementary variables are proportional to the current values is scale invariant. I.e. the dynamics is invariant under rescaling (= a transformation that multiplies all the variables by an arbitrary common factor). The fact that the auto-catalytic dynamics is invariant under rescaling, suggests that it leads to a distribution of the variables which is invariant under rescaling too [8]. The only functions which are invariant under rescaling are the power laws: P(K x) ~ (K x) ^{-1- a}~ x ^{-1- a} ~ P( x).

Note that by taking the logarithm of the variables, random changes proportional to the present value become random additive changes. This brings auto-catalytic dynamics within the realm of statistical mechanics and its powerful methods can be applied efficiently

To get the priorities straight: the power law distribution of word frequencies was discovered Estoup in 1916 [Estoup 1916] long before Zipf) Also the power law in the city size distribution was not discovered by ZZipf but by Auerbach in 1913 [Auerbach 1913]. The power law in the number of papers produced by an author was discovered by Lotka. For other places in information dynamics where power laws appear see [Bookstein 90].

In RNA and proteomic sequences, a very early study was published already in 1955: G Gamow, M Ycas (1955), "Statistical correlation of protein and ribonucleic acid composition", Proceedings of National Academy of Sciences, 41 (12), 1011-1019 (Dec 15, 1955).

The earliest papers connected to power laws in WWW and internet started to appear in the mid 90's: S Glassman (1994), WE Leland et al (1994), C R Cunha, A Bestavros, M E Crovella 1995. V Almeida et al (1996), M F Arlitt, C L Williamson (1997), ME Crovella, A Bestavros (1997), P Barford, ME Crovella , 1997, ME Crovella, M S Taqqu, A Bestavros (1998), N Nishikawa, T Hosokawa, Y Mori, K Yoshida, H Tsuji (1998), BA Huberman, PLT Pirollo, JE Pitkow, RM Lukose, (1998), A-L Barabasi, R Albert (1999).

In the following list, for clarity and brevity, instead of saying:

' The probability density for a person to have wealth x is approximately a power law P(x) ~ x ^{-1- a}, we just make a list entry named 'wealth of an individual' or 'individual wealth':

• Duration of individual stays at one address
• Time a purchaser stays with a supplier
• Duration of browsing a website
• Time for get rid of inventory items
• Time a political party stays in power
• Duration of wars
• Time for device functioning without failure
• Duration patient stays in hospital
• Time to turn prospect into sale.
• Time for searching for missing persons.
• Time for unaccounted teenagers.
• Average survival time of a new business
• Time to complete a painting.
• Time that a bad debt will remain unpaid.
• Duration of engineering projects.
• Assets shares in a Portfolio
• firm size
• Size of rounding error in a final computer result
• Ecological population size
• Bankruptcy sizes.
• Detection of false data by an auditor
• Volume of Website traffic
• Frequencies of words in texts
• The size of human settlements
• File size distribution of Internet traffic
• Clusters of Bose-Einstein condensate
• Size of oil reserves in oil fields
• The length distribution in batched computer jobs
• returns on individual stocks
• Size of sand particles
• Size of meteorites
• Size of moon craters
• Number of species per genus
• Areas burnt in forest fires
• Stored Stock size per product type 
• sales volumes per product type
• profit per customer
• pollution rate per vehicle
• sales results per advertisement
• complaints per product type / service type
• car rentals per customer
• product quantity consumed per consumer
• telephone calls per caller
• frequency of code portion usage
• Decisions per meeting time
• Results per action item
• Number of Interruptions per interrupter
• Occurrences per error type
• Sales per sale-person
• Revenue per company unit
• Amount of Crimes per criminals
• Fruits per plant
• Website / Blog Popularity.
• Search Engine Queries per question
• Distribution of peering sessions per router
• Internet site connections
• Movies-Demand in Video Servers.
• Size Distribution of Firms.
• Territory distribution in a Society.
• income distribution of companies
• Human behavior
• Non-coding portions of DNA.
• size of RNA Structures ,
• Earthquake areas
• Size of Phylogenic tree branches
• Duration of peering sessions carried by routers,
• Size of stored computer files sizes
• the sizes of earthquakes
• size of solar flares
• war duration and intensity,
• the frequency of use of words
• the frequency of occurrence of personal names (in most cultures)
• the number of citations received by papers
• the number of hits on web pages
• the sales of books, CDs
• the numbers of species in biological taxa
• number of calls received by a person
• Frequencies of family names
• Number of protein sequences associated to a protein structure
• Freqences of psychiatric diseases
• heartbeat intervals
• Frequency of family names
• Nr of Species with individuals of a given size
• Nr of Species vs number of specimens
• Nr of Species vs their life time
• Nr of Languages vs number of speakers
• Nr of countries vs population / size
• Nr of towns vs. population
• Nr of product types vs. number of units sold
• Nr of treatments vs number of patients treated
• Nr of patients vs cost of treatment
• Nr of moon craters vs their size
• Nr of earthquakes vs their strenth
• Nr of meteorites vs their size
• Nr of voids vs their size
• Nr of galaxies vs their size
• Nr of rives vs the size of their basin

A promising concept which might dominate this direction for the coming years is stochastic logistic systems of generalized Lotka-Volterra type [9] (spatially distributed logistic) with random coefficients r_{ij .}

Next: Multi-Agent Modeling of Reaction-Diffusion Systems
Up: Table of contents
Previous: Logistic Systems