Complex systems 2
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1. The document discusses scaling and self-similarity in complex systems, including Zipf's law and Pareto distributions. 2. It introduces Zipf's law which shows a power law relationship between variables on a log-log plot, and Pareto distributions which follow a power law pattern seen in many natural phenomena. 3. Scaling and self-similarity are properties of systems where the patterns remain similar across scales, seen in fractals and other complex systems from cities to earthquakes that can be modeled with power laws.
Complex systems 2
- 1. Complex Systems Session 2. Pareto distribution, long-tails, scaling and self-similarity
- 2. Zipf’s law SzEEDSM Complex Systems 2017
- 3. George K. Zipf (1902-1950) SzEEDSM Complex Systems 2017 Rank-order plot Zipf’s law (1941)
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- 5. Log-log plot A power law relationship between two variables yields a linear relationship between the logarithms of the quantities. If we plot the quantities on a log-log plot we should see a line. SzEEDSM Complex Systems 2017
- 6. Zipf’s law SzEEDSM Complex Systems 2017
- 7. City size distribution SzEEDSM Complex Systems 2017
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- 12. Vilfredo Pareto (1848-1923) SzEEDSM Complex Systems 2017
- 13. Pareto principle 80/20 rule Italian land ownership British tax records Peapods in Pareto’s garden SzEEDSM Complex Systems 2017
- 14. Pareto distribution Tail distribution Power law tail Long tail Fat tail α= 1+ ε epsilon is a symbol of a small number SzEEDSM Complex Systems 2017
- 15. The 80/20 rule SzEEDSM Complex Systems 2017
- 16. The 80/20 rule SzEEDSM Complex Systems 2017
- 17. Inequality and Gini coefficient SzEEDSM Complex Systems 2017
- 18. Normal distribution SzEEDSM Complex Systems 2017
- 19. Complementary cumulative distribution or Tail Distribution SzEEDSM Complex Systems 2017
- 20. The Power Law Distribution Power law distributions are ubiquitous, occurring in diverse phenomena, including city sizes, incomes, word frequencies, and earthquake magnitudes. Power laws easy to spot on log-log (doubly logarithmic) plots: Power Law PDF - Linear Scale Power Law PDF – Log-Log scale α=0.5 α=1.0 α=2.0
- 21. Scale invariance and self-similarity SzEEDSM Complex Systems 2017
- 22. Scale invariance Pareto: distribution of wealth exceeding a minimum xm Change the SCALE and take those, who exceed xt Pareto remains the same on the new SCALE x/xt There is a lack of natural SCALE in the problem Only Power Laws are invariant under change of scale SzEEDSM Complex Systems 2017
- 23. Scaling (scale invariance) Distribution of height of people is NOT scale invariant. Humans do have a typical height scale (1-2 meters). There are no 10 m tall humans. Tallness is not power law distributed Number of people you can relate to (Dunbar number) 150 Number of years you can live (no one lived for 150 years) Number of followers 1,10,100,1000,…,10 million Number of sexual partners 1,10,100,1000 ... Attraction, hype, fanship ... SzEEDSM Complex Systems 2017
- 24. Self-similar objects: Fractals SzEEDSM Complex Systems 2017
- 25. Benoit Mandelbrot (1924-2010) SzEEDSM Complex Systems 2017
- 26. Mandelbrot set Z2+C SzEEDSM Complex Systems 2017
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- 28. SzEEDSM Complex Systems 2017
- 29. Internet vs. Phone SzEEDSM Complex Systems 2017
- 30. Lorenz Attractor SzEEDSM Complex Systems 2017
- 31. The Universe SzEEDSM Complex Systems 2017
- 32. Coastline of Britain SzEEDSM Complex Systems 2017
- 33. Barnsley Fern SzEEDSM Complex Systems 2017
- 34. Fractal dimension SzEEDSM Complex Systems 2017
- 35. Fractal Dimension SzEEDSM Complex Systems 2017
- 36. Urban Scaling SzEEDSM Complex Systems 2017
- 37. SzEEDSM Complex Systems 2017
- 38. Metabolic rate of mammals SzEEDSM Complex Systems 2017
- 39. Road length vs. GDP SzEEDSM Complex Systems 2017
- 40. Scaling vs. complexity SzEEDSM Complex Systems 2017
- 41. Scaling and Politics SzEEDSM Complex Systems 2017
- 42. Universality Biology City Land ownership Japanese company sales Retail industry Earthquakes … SzEEDSM Complex Systems 2017
Editor's Notes
- #21: Zipf was a Harvard linguist interested in the frequency of word usage versus the ranked list of common words. Pareto studied distribution of wealth and income (cumulative wealth greater than x). Sometimes also called the 80/20 rule (20% of the people own 80% of the wealth).