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

4. Example: Population of cities follows a power-law (“scale-free) distribution
2006/09/350px_US_Metro_popultion_graph.png

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

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. A power law, plotted on a “log-log” plot, is a straight line.
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

9. Other examples of power laws in nature Gutenberg-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. (

14. Rank-frequency scaling: City populations

15. Rank-frequency scaling: Income distribution

16. From A Unified Theory of Urban Living, L. Bettencourt and G
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

18. What causes these distributions?

19. Interesting distribution: “Benford’s law”

20. In-class exercise: Benford’s Law
City populations

21. Benford’s law: Distribution of leading digits
Newcomb’s observation
Explanation of Benford’s law?

22. Plot deviations from Benford’s law versus year
Collect distribution of leading digits in corporate accounting statements of total assets
Plot deviations from Benford’s law versus year

23. “Bernie vs Benford’s Law: Madoff Wasn’t That Dumb”
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. Surface area of a sphere:
Volume of a sphere:
Surface area of a sphere:
Surface area scales with volume to the 2/3 power.
“Volume of a sphere scales as the radius cubed”
“Surface area of a sphere scales as the
radius squared”
mouse
hamster
(8  mouse mass)
hippo
(125,000  mouse mass)

34. Surface area of a sphere:
Volume of a sphere:
Surface area of a sphere:
Surface area scales with volume to the 2/3 power.
“Volume of a sphere scales as the radius cubed”
“Surface area of a sphere scales as the
radius squared”
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: Hypothesis 3 (“Keiber’s law): metabolic rate  mass3/4
y = x3/4
Hypothesis 3 (“Keiber’s law): metabolic rate  mass3/4

38. For sixty years, no explanation
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 = x 2/3 y = x 3/4 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.

51. “Although living things occupy a three-dimensional space, their internal physiology and anatomy operate as if they were four-dimensional. . . Fractal geometry has literally given life an added dimension.”
― West, Brown, and Enquist

52. Critiques of their model
E.g.,
Bottom line: Model is interesting and elegant, but both the explanation and the underlying data are controversial.
Validity of these ideas beyond biology?

53. Do fractal distribution networks explain scaling in cities. Cf
Do fractal distribution networks explain scaling in cities? Cf. Bettencourt, Lobo, Helbing, Kuhnert, and West, PNAS 2007
“[L]ife at all scales is sustained by optimized, space- filling, hierarchical branching networks, which grow with the size of the organism as uniquely specified approximately self-similar structures.”

54. Total wages per metropolitan area vs. population

55. Walking speed vs. population

56. “Supercreative” employment vs. population