Networks and Scaling

1. Networks and Scaling

2. Distributions and Scaling • What is a numerical distribution? • What is scaling?

3. Example: Human height follows a normal distribution Frequency Height http://scienceblogs.com/builtonfacts/2009/02/the_central_limit_theorem_made.php

4. Example: Population of cities follows a power-law (“scale-free) distribution http://upload.wikimedia.org/wikipedia/commons/4/49/Powercitiesrp.png http://www.streetsblog.org/wp-content/uploads 2006/09/350px_US_Metro_popultion_graph.png http://cheapukferries.files.wordpress.com/2010/06/hollandcitypopulation1.png

5. part of WWW Number of nodes Number of nodes Degree Degree

6. The Web’s approximate Degree Distribution “Scale-free” distribution Number of nodes “power law” Degree

7. The Web’s approximate Degree Distribution “Scale-free” distribution Number of nodes “power law” Degree

8. log (Number of nodes) Number of nodes Degree k log (Degree) A power law, plotted on a “log-log” plot, is a straight line. The slope of the line is the exponent of the power law. From http://www.pnas.org/content/105/37/13724/F4.expansion.html

9. Other examples of power laws in natureGutenberg-Richter law of earthquake magnitudes By: Bak [1]

10. Metabolic scaling in animals

11. Rank-frequency scaling: Word frequency in English (Zipf’s law) A plot of word frequency of single words (unigrams) versus rank r extracted from the one million words of the Brown’s English dictionary. (http://web.me.com/kristofferrypdal/Themes_Site/Scale_invariance.html)

12. http://cs.pervasive.com/blogs/datarush/Figure2.png

14. Rank-frequency scaling: City populations http://brenocon.com/blog/2009/05/zipfs-law-and-world-city-populations/

15. Rank-frequency scaling: Income distribution

16. From A Unified Theory of Urban Living, L. Bettencourt and G. West, Nature, 467, 912–913, 2010 Scaling in cities

17. http://mjperry.blogspot.com/2008/08/more-on-medal-inequality-at-2008.htmlhttp://mjperry.blogspot.com/2008/08/more-on-medal-inequality-at-2008.html

18. What causes these distributions?

19. Interesting distribution: “Benford’s law”

20. In-class exercise: Benford’s Law • City populations http://www.census.gov/population/www/documentation/twps0027/tab22.txt

21. Benford’s law: Distribution of leading digits http://www.youtube.com/watch?v=O8N26edbqLM Newcomb’s observation Explanation of Benford’s law?

22. Collect distribution of leading digits in corporate accounting statements of total assets Plot deviations from Benford’s law versus year http://econerdfood.blogspot.com/2011/10/benfords-law-and-decreasing-reliability.html

23. “Bernie vsBenford’s Law: Madoff Wasn’t That Dumb” http://paul.kedrosky.com/archives/2008/12/bernie_vs_benfo.html Frequency of leading digits in returns reported by Bernie Madoff’s funds

24. Controversy: Can Network Structure and Dynamics Explain Scaling in Biology and Other Disciplines? Scaling: How do properties of systems (organisms, economies, cities) change as their size is varied? Example: How does basal metabolic rate (heat radiation) vary as a function of an animal’s body mass?

25. Metabolic scaling • Surface hypothesis: • Body is made of cells, in which metabolic reactions take place. • Can “approximate” body mass by a sphere of cells with radius r. • Can approximate metabolic rate by surface area r

26. Mouse Hamster Hippo

27. Mouse Hamster Radius = 2  Mouse radius Hippo Radius = 50  Mouse radius

28. Mouse Hamster Radius = 2  Mouse radius Hippo Radius = 50  Mouse radius Hypothesis 1: metabolic rate  body mass

29. Mouse Hamster Radius = 2  Mouse radius Hippo Radius = 50  Mouse radius Hypothesis 1: metabolic rate  body mass Problem: Mass is proportional to volume of animal but heat can radiate only from surface of animal

30. Mouse Hamster Radius = 2  Mouse radius Volume of a sphere: Surface area of a sphere: Hippo Radius = 50  Mouse radius Hypothesis 1: metabolic rate  body mass Problem: mass is proportional to volume of animal but heat can radiate only from surface of animal

31. Mouse Hamster Radius = 2  Mouse radius Mass 8  Mouse radius Surface area 4  Mouse radius Volume of a sphere: Surface area of a sphere: Hippo Radius = 50  Mouse radius Hypothesis 1: metabolic rate  body mass Problem: mass is proportional to volume of animal but heat can radiate only from surface of animal

32. Mouse Hamster Radius = 2  Mouse radius Mass 8  Mouse radius Surface area 4  Mouse radius Volume of a sphere: Surface area of a sphere: Hippo Radius = 50  Mouse radius Mass 125,000  Mouse radius Surface area  2,500  Mouse radius Hypothesis 1: metabolic rate  body mass Problem: mass is proportional to volume of animal but heat can radiate only from surface of animal

33. “Volume of a sphere scales as the radius cubed” “Surface area of a sphere scales as the radius squared” Volume of a sphere: Surface area of a sphere: Surface area scales with volume to the 2/3 power. mouse hamster (8  mouse mass) hippo (125,000  mouse mass)

34. “Volume of a sphere scales as the radius cubed” “Surface area of a sphere scales as the radius squared” Volume of a sphere: Surface area of a sphere: Surface area scales with volume to the 2/3 power. Hypothesis 2 (“Surface Hypothesis): metabolic rate mass2/3 mouse hamster (8  mouse mass) hippo (125,000  mouse mass)

35. y = x2/3 log (metabolic rate) log (body mass)

36. Actual data: y = x3/4

37. Actual data: y = x3/4 Hypothesis 3 (“Keiber’s law): metabolic rate mass3/4

38. Actual data: y = x3/4 Hypothesis 3 (“Keiber’s law): metabolic rate mass3/4 For sixty years, no explanation

39. Kleiber’s law extended over 21 orders of magnitude

40. y = x2/3 y = x3/4 More “efficient”, in sense that metabolic rate (and thus rate of distribution of nutrients to cells) is larger than surface area would predict. metabolic rate body mass

41. Other Observed Biological Scaling Laws Heart rate  body mass1/4 Blood circulation time  body mass1/4 Life span  body mass1/4 Growth rate  body mass1/4 Heights of trees  tree mass1/4 Sap circulation time in trees  tree mass1/4

42. West, Brown, and Enquist’s Theory(1990s)

43. West, Brown, and Enquist’s Theory(1990s) General idea: “metabolic scaling rates (and other biological rates) are limited not by surface area but by rates at which energy and materials can be distributed between surfaces where they are exchanged and the tissues where they are used. “ How are energy and materials distributed?

44. Distribution systems

46. West, Brown, and Enquist’s Theory(1990s) • Assumptions about distribution network: • branches to reach all parts of three-dimensional organism (i.e., needs to be as “space-filling” as possible) • has terminal units (e.g., capillaries) that do not vary with size among organisms • evolved to minimize total energy required to distribution resources

49. Prediction: Distribution network will have fractal branching structure, and will be similar in all / most organisms (i.e., evolution did not optimize distribution networks of each species independently) • Therefore, Euclidean geometry is the wrong way to view scaling; one should use fractal geometry instead! • With detailed mathematical model using three assumptions, they derive metabolic rate body mass3/4

50. Their interpretation of their model • Metabolic rate scales with body mass like surface area scales with volume... but in four dimensions.