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Network Thinking

Complexity: A Guided Tour, Chapters 15-16

Neural Network (C. Elegans)

http://gephi.org/wp-content/uploads/2008/12/screenshot-celegans.png

Food Web

http://1.bp.blogspot.com/_vIFBm3t8boU/SBhzqbchIeI/AAAAAAAAAXk/RsC-Pj45Avc/s400/food%2Bweb.bmp

Metabolic Network

http://www.funpecrp.com.br/gmr/year2005/vol3-4/wob01_full_text.htm

Genetic Regulatory Network

http://expertvoices.nsdl.org/cornell-info204/files/2009/03/figure-3.jpeg

Bank Network

From Schweitzer et al., Science, 325, 422-425, 2009 http://www.sciencemag.org/cgi/content/full/325/5939/422

Airline Routes

http://virtualskies.arc.nasa.gov/research/tutorial/images/12routemap.gif

US Power Grid

http://images.encarta.msn.com/xrefmedia/aencmed/targets/maps/map/000a5302.gif

Internet

http://www.visualcomplexity.com/vc/images/270_big01.jpg

World Wide Web (small part)

From M. E. J. Newman and M. Girvin, Physical Review Letters E, 69, 026113, 2004.

Social Network

http://ucsdnews.ucsd.edu/graphics/images/2007/07-07socialnetworkmapLG.jpg

The Science of Networks

Are there properties common to all complex networks?



If so, why?



If so, why?

Can we formulate a general theory of the structure, evolution, and dynamics of networks?


Small-World Property (Watts and Strogatz, 1998)


me


me

Barack Obama


me

Barack Obama

my mother


me

Barack Obama

my mother

Nancy Bekavac


me

Barack Obama

my mother

Nancy Bekavac

Hillary Clinton


me

Barack Obama


me my cousin Matt Dunne

Barack Obama


me

Barack Obama

Patrick Leahy

my cousin Matt Dunne

Stanley Milgram

Stanley Milgram

Nebraska farmer

Boston stockbroker

Stanley Milgram

On average: “six degrees of separation”

Nebraska farmer

Boston stockbroker

The Small-World Property

The network has relatively few “long-distance”

links but there are short paths between most

pairs of nodes, usually created by “hubs”.




Most real-world complex networks seem to

have the small-world property!





Most real-world complex networks seem to

have the small-world property!

But why?


And how can the shortest paths actually be

found?


http://oracleofbacon.org/

Six Degrees of Kevin Bacon

Measure the average distance between Kevin Bacon and all other actors.

No. of movies : 46 No. of actors : 1811 Average separation: 2.79 Kevin Bacon

Is Kevin Bacon the most connected actor?

NO!

876 Kevin Bacon 2.786981 46 1811

From http://www.dmae.upm.esWebpersonalBartolo/

•  Degree •  Number of edges connected to a node.

•  In-degree •  Number of incoming edges.

•  Out-degree •  Number of outgoing edges.


Network parameters

Diameter Maximum distance between any pair of nodes.

Path length: number of “hops” to get from node v1 to node v2

Connectivity Number of neighbors of a given node: k := degree.

P(k) := Probability of having k neighbors.

Clustering Are neighbors of a node also neighbors among them?


Clustering coefficient of a node v

C(v) = # of links between neighbors n(n-1)/2

Clustering: My friends will know each other with high probability! (typical example: social networks)

C(v) = 4/6

C is the average over all C(v)


Real life networks are clustered, large C, but have small average distance L.

Duncan J. Watts & Steven H. Strogatz, Nature 393, 440-442 (1998)

L L rand C C rand N WWW 3.1 3.35 0.11 0.00023 153127 Actors 3.65 2.99 0.79 0.00027 225226 Power Grid 18.7 12.4 0.080 0.005 4914 C. Elegans 2.65 2.25 0.28 0.05 282


Select a fraction p of edges Reposition on of their endpoints

Watts-Strogatz model: Generating small world graphs

Source: Watts, D.J., Strogatz, S.H.(1998) Collective dynamics of 'small-world' networks. Nature 393:440-442.

From www.cse.unr.edu/~mgunes/cs765/cs790f10/Lect18_SmallWorld.ppt

Netlogo: Small Worlds

•  Each node has K>=4 nearest neighbors (local)

•  tunable: vary the probability p of rewiring any given edge

•  small p: regular lattice

•  large p: classical random graph

Watts-Strogatz model: Generating small world graphs


Watts/Strogatz model: What happens in between?

•  Small shortest path means small clustering?

•  Large shortest path means large clustering?

•  Through numerical simulation –  As we increase p from 0 to 1

•  Fast decrease of mean distance •  Slow decrease in clustering


Watts/Strogatz model: Change in clustering coefficient and average path length as a function of

the proportion of rewired edges

l(p)/l(0)

C(p)/C(0)

10% of links rewired 1% of links rewired Source: Watts, D.J., Strogatz, S.H.(1998) Collective dynamics of 'small-world' networks. Nature 393:440-442.


Structured network •  high clustering •  large diameter •  regular

Random network •  small clustering •  small diameter

Small-world network •  high clustering •  small diameter •  almost regular

N = 1000 k =10 D = 100 L = 49.51 C = 0.67

N =1000 k = 8-13 D = 14 d = 11.1 C = 0.63

N =1000 k = 5-18 D = 5 L = 4.46 C = 0.01


Scale-Free Structure (Albert and Barabási, 1998)

Scale-Free Structure (Albert and Barabási, 1998)

Typical structure of

World Wide Web (nodes = web pages, links = links between pages)

Typical structure of

a randomly connected

network http://www.dichotomistic.com/images/random%20network.gif

part of WWW

Concept of “Degree Distribution”

A node with degree 3

Concept of “Degree Distribution”

1 2 3 4 5 6 7 8 9 10

Degree

Number of Nodes

6

5

4

3

2

1

0

A node with degree 3

part of WWW

Degree

Num

ber o

f nod

es

Degree

Num

ber o

f nod

es

The Web’s approximate Degree Distribution N

umbe

r of n

odes

N

umbe

r of n

odes

Degree

Num

ber o

f nod

es

Num

ber o

f nod

es

Degree

The Web’s approximate Degree Distribution

Num

ber o

f nod

es


umbe

r of n

odes

Degree

Degree


umbe

r of n

odes

Degree

“Scale-free” distribution


umbe

r of n

odes

Degree


€

Number of nodes with degree k ≈ 1k 2


umbe

r of n

odes

Degree



umbe

r of n

odes

“power law”

Degree



umbe

r of n

odes

“power law”

“Scale-free” distribution = “power law” distribution

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

The scale-free structure of the Web helps to explain why Google works so well

The scale-free structure of the Web helps to explain why Google works so well

It also explains some of the success of other scale-free networks in nature!

part of WWW

Scale-Free Networks are “fractal-like”

http://en.wikipedia.org/wiki/File:WorldWideWebAroundGoogle.png

Scale-Free Networks have high clustering

part of WWW High Clustering:

Low Clustering:

High-Clustering Helps in Discovering Community Structure in Networks

How are Scale-Free Networks Created?

Web pages

Preferential attachment demo

(Netlogo)

Robustness of Scale-Free Networks


•  Vulnerable to targeted “hub” failure



•  Robust to random node failure



•  Robust to random node failure

unless....

nodes can cause other nodes to fail

Can result in cascading failure

August, 2003 electrical blackout in northeast US and Canada

9:29pm 1 day before

9:14pm Day of blackout

http://earthobservatory.nasa.gov/ images/imagerecords/3000/3719/ NE_US_OLS2003227.jpg

http://www.geocities.com/WallStreet/Exchange/9807/Charts/SP500/fdicfail_0907.jpg

We see similar patterns of cascading failure in

biological systems, ecological systems, computer and

communication networks, wars, etc.

Normal (“bell-curve) distribution

http://resources.esri.com/help/9.3/arcgisdesktop/com/gp_toolref/process_simulations_sensitivity_analysis_and_error_analysis_modeling/Random_Normal_Distribution.gif

Normal (“bell-curve) distribution

http://resources.esri.com/help/9.3/arcgisdesktop/com/gp_toolref/process_simulations_sensitivity_analysis_and_error_analysis_modeling/Random_Normal_Distribution.gif

“Events in ‘tail’ are highly unlikely”

Power law (“scale free”) distribution

http://www.marketoracle.co.uk/images/mauldin_16_10_07image003.gif



Notion of “heavy tail”:

Events in tail are more likely than in normal distribution


“More normal than ‘normal’ ”



“More normal than ‘normal’ ”


“Few economists saw our current crisis coming, but this predictive failure was the least of the field’s problems. More important was the profession’s blindness to the very possibility of catastrophic failures in a market economy.

-- Paul Krugman, New York Times, September 6, 2009

Observed common properties:

•  Small world property

•  Scale-free degree distribution

•  Clustering and community structure

•  Robustness to random node failure

•  Vulnerability to targeted hub attacks

•  Vulnerability to cascading failures

Other examples of power-laws in nature

•  Magnitude vs. frequency of earthquakes

•  Magnitude vs. frequency of stock market crashes

•  Income vs. frequency (of people with that income)

•  Populations of cities vs. frequency (of cities with that population)

•  Word rank vs. frequency in English text

•  Binomial distribution demo:

http://www.bhsstatistics.com/Applets/Applets/binomialdemo1.html

http://www.jcu.edu/math/isep/Quincunx/Quincunx.html

•  Sandpile demo http://www.visualentities.com/sandpile.htm

What does a power law distribution look like on a logarithmic plot, and why?

Gutenberg-Richter Law

By: Bak [1]

Regularity of Biological Extinctions

By: Bak [1]

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