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Networks and Scaling

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

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Example: Human height follows a normal distribution Height Frequency

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Example: Population of cities follows a power-law (“scale- free) distribution /09/350px_US_Metro_popultion_graph.png citypopulation1.png

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part of WWW Degree Number of nodes Degree Number of nodes

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

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

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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

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

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Metabolic scaling in animals

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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)

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Rank-frequency scaling: City populations

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Rank-frequency scaling: Income distribution

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

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What causes these distributions?

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Interesting distribution: “Benford’s law”

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In-class exercise: Benford’s Law City populations

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

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

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

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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?

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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

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Mouse Hamster Hippo

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

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

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

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

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

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

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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)

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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) Hypothesis 2 (“Surface Hypothesis): metabolic rate mass 2/3

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y = x 2/3 log (body mass) log (metabolic rate)

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Actual data: y = x 3/4

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Actual data: Hypothesis 3 (“Keiber’s law): metabolic rate mass 3/4 y = x 3/4

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

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Kleiber’s law extended over 21 orders of magnitude

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y = x 3/4 y = x 2/3 metabolic rate body mass More “efficient”, in sense that metabolic rate (and thus rate of distribution of nutrients to cells) is larger than surface area would predict.

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

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

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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?

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Distribution systems

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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

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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 mass 3/4

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

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“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

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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?

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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.”

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Total wages per metropolitan area vs. population

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Walking speed vs. population

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“Supercreative” employment vs. population

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