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
Example: Population of cities follows a power-law (“scalefree) 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/holland
citypopulation1.png
Number of nodes
Number of nodes
part of WWW
Degree
Degree
The Web’s approximate Degree Distribution
Number of nodes
“Scale-free” distribution
Number of nodes with degree k 
“power law”
Degree
1
k2
The Web’s approximate Degree Distribution
Number of nodes
“Scale-free” distribution
Number of nodes with degree k 
“power law”
Degree
1
k2
log (Number of nodes)
Number of nodes

Degree k
log (Degree)
1
k 2
k
 1 
log k  log  2  log k 2   2log k
k 

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
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 populations
http://www.census.gov/population/www/documentation/twps0027/tab22.txt
Benford’s law: Distribution of leading digits
http://www.youtube.com/watch?v=O8N26edbqLM
Newcomb’s observation
Explanation of Benford’s law?
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
Hamster
Radius = 2  Mouse radius
Hippo
Radius = 50  Mouse radius
Mouse
Hamster
Radius = 2  Mouse radius
Hypothesis 1: metabolic rate  body
mass
Hippo
Radius = 50  Mouse radius
Mouse
Hamster
Radius = 2  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
Hippo
Radius = 50  Mouse radius
Mouse
Hamster
Radius = 2  Mouse radius
Volume of a sphere:
4 3
r
3
Surface area of a sphere: 4 r 2
Hypothesis 1: metabolic rate  body
mass
Problem:
mass is proportional to volume of animal
but heat can radiate only from surface of animal
Hippo
Radius = 50  Mouse radius
Mouse
Hamster
Radius = 2  Mouse radius
Mass  8  Mouse radius
Surface area  4  Mouse radius
Volume of a sphere:
4 3
r
3
Surface area of a sphere: 4 r 2
Hypothesis 1: metabolic rate  body
mass
Problem:
mass is proportional to volume of animal
but heat can radiate only from surface of animal
Hippo
Radius = 50  Mouse radius
Mouse
Hamster
Radius = 2  Mouse radius
Mass  8  Mouse radius
Surface area  4  Mouse radius
Volume of a sphere:
4 3
r
3
Surface area of a sphere: 4 r 2
Hypothesis 1: metabolic rate  body
mass
Problem:
mass is proportional to volume of animal
but heat can radiate only from surface of animal
Hippo
Radius = 50  Mouse radius
Mass  125,000
 Mouse radius
Surface area 
2,500  Mouse radius
Volume of a sphere:
4 3
r
3
Surface area of a sphere:
“Volume of a sphere scales as the radius cubed”
4 r 2 “Surface area of a sphere scales as the
radius squared”
Surface area scales with volume to the 2/3 power.
mouse
hamster
(8  mouse mass)
hippo
(125,000  mouse mass)
Volume of a sphere:
4 3
r
3
Surface area of a sphere:
“Volume of a sphere scales as the radius cubed”
4 r 2 “Surface area of a sphere scales as the
radius squared”
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)
y = x2/3
log (metabolic rate)
log (body mass)
Actual data:
y = x3/4
Actual data:
y = x3/4
Hypothesis 3 (“Keiber’s law): metabolic rate  mass3/4
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. Bettencourt, Lobo, Helbing, Kuhnert,
and West, PNAS 2007
“[L]ife at all scales is sustained by optimized, spacefilling, 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