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# Networks and Scaling.

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Networks and Scaling

Distributions and Scaling
What is a numerical distribution? What is scaling?

Example: Human height follows a normal distribution
Frequency Height

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

part of WWW Degree Number of nodes Degree Number of nodes

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

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

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

Other examples of power laws in nature Gutenberg-Richter law of earthquake magnitudes
By: Bak [1]

Metabolic scaling in animals

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)

Rank-frequency scaling: City populations

Rank-frequency scaling: Income distribution

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

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

What causes these distributions?

Interesting distribution: “Benford’s law”

In-class exercise: Benford’s Law
City populations

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

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

“Bernie vs Benford’s Law: Madoff Wasn’t That Dumb”
Frequency of leading digits in returns reported by Bernie Madoff’s funds

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?

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

Mouse Hamster Hippo

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

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

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

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

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

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

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)

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)

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

Actual data: y = x3/4

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

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

Kleiber’s law extended over 21 orders of magnitude

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

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

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

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?

Distribution systems

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

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

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

“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

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?

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.”

Total wages per metropolitan area vs. population

Walking speed vs. population

“Supercreative” employment vs. population

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