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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. http://scienceblogs.com/builtonfacts/2009/02/the_central_limit_theorem_made.php. - PowerPoint PPT Presentation
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Networks and Scaling
Distributions and Scaling
• What is a numerical distribution?
• What is scaling?
http://scienceblogs.com/builtonfacts/2009/02/the_central_limit_theorem_made.php
Example: Human height follows a normal distribution
Height
Frequency
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
part of WWW
Degree
Num
ber o
f nod
es
Degree
Num
ber o
f nod
es
Degree
“Scale-free” distribution
21
k kdegreewithnodesofNumber
The Web’s approximate Degree Distribution
Num
ber o
f nod
es
“power law”
Degree
“Scale-free” distribution
21
k kdegreewithnodesofNumber
The Web’s approximate Degree Distribution
Num
ber o
f nod
es
“power law”
€
k ≈1k 2
log
(Num
ber o
f nod
es)
Num
ber o
f nod
es
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
€
logk ≈ log1k 2
⎛ ⎝ ⎜
⎞ ⎠ ⎟= log k −2( ) = −2logk
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)
http://cs.pervasive.com/blogs/datarush/Figure2.png
Rank-frequency scaling: City populations
http://brenocon.com/blog/2009/05/zipfs-law-and-world-city-populations/
Rank-frequency scaling: Income distribution
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.html
What causes these distributions?
Interesting distribution: “Benford’s law”
In-class exercise: Benford’s Law
• City populationshttp://www.census.gov/population/www/documentation/twps0027/tab22.txt
Benford’s law: Distribution of leading digits
Newcomb’s observation
Explanation of Benford’s law?
http://www.youtube.com/watch?v=O8N26edbqLM
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
“Bernie vs Benford’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
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
HamsterRadius = 2 Mouse radius
HippoRadius = 50 Mouse radius
Mouse
HamsterRadius = 2 Mouse radius
HippoRadius = 50 Mouse radiusHypothesis 1: metabolic rate body
mass
Problem: Mass is proportional to volume of animalbut heat can radiate only from surface of animal
Mouse
HamsterRadius = 2 Mouse radius
Hypothesis 1: metabolic rate body mass
HippoRadius = 50 Mouse radius
Problem: mass is proportional to volume of animalbut heat can radiate only from surface of animal
Mouse
HamsterRadius = 2 Mouse radius
Hypothesis 1: metabolic rate body mass
HippoRadius = 50 Mouse radius
Volume of a sphere:
Surface area of a sphere:
3
34 r
24 r
Problem: mass is proportional to volume of animalbut heat can radiate only from surface of animal
Mouse
Hypothesis 1: metabolic rate body mass
HippoRadius = 50 Mouse radius
Volume of a sphere:
Surface area of a sphere:
3
34 r
24 r
HamsterRadius = 2 Mouse radiusMass 8 Mouse radiusSurface area 4 Mouse radius
Problem: mass is proportional to volume of animalbut heat can radiate only from surface of animal
Mouse
Hypothesis 1: metabolic rate body mass
Volume of a sphere:
Surface area of a sphere:
3
34 r
24 r
HamsterRadius = 2 Mouse radiusMass 8 Mouse radiusSurface area 4 Mouse radius
HippoRadius = 50 Mouse radiusMass 125,000 Mouse radiusSurface area 2,500 Mouse radius
Volume of a sphere:
Surface area of a sphere:
Surface area scales with volume to the 2/3 power.
3
34 r
24 r
“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)
Volume of a sphere:
Surface area of a sphere:
Surface area scales with volume to the 2/3 power.
3
34 r
24 r
“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 mass2/3
y = x2/3
log (body mass)
log (metabolic rate)
Actual data: y = x3/4
Actual data:
Hypothesis 3 (“Keiber’s law): metabolic rate mass3/4
y = x3/4
Actual data:
For sixty years, no explanation
Hypothesis 3 (“Keiber’s law): metabolic rate mass3/4
y = x3/4
Kleiber’s law extended over 21 orders of magnitude
y = x 3/4
y = x 2/3
metabolicrate
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.
Other Observed Biological Scaling Laws
Heart rate body mass1/4
Blood circulation time body mass1/4
Life span body mass1/4
Growth rate body mass1/4
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 explainscaling 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
http://www.youtube.com/watch?v=SI9H4lECB6E
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