Networks and Scaling

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

http://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


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


Hypothesis 1: metabolic rate body mass


Problem: mass is proportional to volume of animalbut heat can radiate only from surface of animal

Mouse




Volume of a sphere:

Surface area of a sphere:

3

34 r

24 r


Mouse



Volume of a sphere:


3

34 r

24 r

HamsterRadius = 2 Mouse radiusMass 8 Mouse radiusSurface area 4 Mouse radius


Mouse


Volume 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 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 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)


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


• 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




Documents

Networks and Scaling