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/RsCPj45Avc/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
The Science of Networks
Are there properties common to all complex
networks?
The Science of Networks
Are there properties common to all complex
networks?
If so, why?
The Science of Networks
Are there properties common to all complex
networks?
If so, why?
Can we formulate a general theory of the structure,
evolution, and dynamics of networks?
Small-World Property
(Watts and Strogatz, 1998)
Small-World Property
(Watts and Strogatz, 1998)
Small-World Property
(Watts and Strogatz, 1998)
me
Small-World Property
(Watts and Strogatz, 1998)
me
Barack
Obama
Small-World Property
(Watts and Strogatz, 1998)
me
Barack
Obama
my mother
Small-World Property
(Watts and Strogatz, 1998)
me
Nancy Bekavac
Barack
Obama
my mother
Small-World Property
(Watts and Strogatz, 1998)
me
Hillary
Clinton
Nancy Bekavac
Barack
Obama
my mother
Small-World Property
(Watts and Strogatz, 1998)
me
Hillary
Clinton
Nancy Bekavac
Barack
Obama
my mother
Small-World Property
(Watts and Strogatz, 1998)
me
Barack
Obama
Small-World Property
(Watts and Strogatz, 1998)
me
my cousin Matt Dunne
Barack
Obama
Small-World Property
(Watts and Strogatz, 1998)
me
my cousin Matt Dunne
Patrick
Leahy
Barack
Obama
Small-World Property
(Watts and Strogatz, 1998)
me
my cousin Matt Dunne
Patrick
Leahy
Barack
Obama
Stanley Milgram
Nebraska farmer
Boston stockbroker
Stanley Milgram
Nebraska farmer
Boston stockbroker
Stanley Milgram
Nebraska farmer
Boston stockbroker
Stanley Milgram
Nebraska farmer
Boston stockbroker
Stanley Milgram
On average: “six degrees of
separation”
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”.
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!
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!
But why?
The Small-World Property
And how can the shortest paths actually be
found?
Six Degrees of Kevin Bacon
http://oracleofbacon.org/
From http://www.dmae.upm.esWebpersonalBartolo/
Measure the average distance between Kevin Bacon and all other actors.
Kevin Bacon
Is Kevin Bacon
the most
connected actor?
NO!
No. of movies : 46
No. of actors : 1811
Average separation: 2.79
Rod Steiger
Donald Pleasence
Martin Sheen
Christopher Lee
Robert Mitchum
Charlton Heston
Eddie Albert
Robert Vaughn
Donald Sutherland
John Gielgud
Anthony Quinn
James Earl Jones
Average
distance
2.537527
2.542376
2.551210
2.552497
2.557181
2.566284
2.567036
2.570193
2.577880
2.578980
2.579750
2.584440
# of
movies
112
180
136
201
136
104
112
126
107
122
146
112
# of
links
2562
2874
3501
2993
2905
2552
3333
2761
2865
2942
2978
3787
Kevin Bacon
Kevin
Bacon
2.786981
2.786981
46
46
1811
1811
Rank
Name
1
2
3
4
5
6
7
8
9
10
11
12
…
876
876
…
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.
From http://www.dmae.upm.esWebpersonalBartolo/
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?
From http://www.dmae.upm.esWebpersonalBartolo/
Clustering coefficient of a node v
# of links between neighbors
C(v) =
C(v) = 4/6
n(n-1)/2
C is the average over all C(v)
Clustering: My friends will know each other with high probability!
(typical example: social networks)
From http://www.dmae.upm.esWebpersonalBartolo/
Duncan J. Watts & Steven H. Strogatz,
Nature 393, 440-442 (1998)
Real life networks are clustered, large C, but have small
average distance L.
WWW
Actors
Power Grid
C. Elegans
L
3.1
3.65
18.7
2.65
Lrand
3.35
2.99
12.4
2.25
C
0.11
0.79
0.080
0.28
Crand
0.00023
0.00027
0.005
0.05
N
153127
225226
4914
282
From www.cse.unr.edu/~mgunes/cs765/cs790f10/Lect18_SmallWorld.ppt
Watts-Strogatz model:
Generating small world graphs
Select a fraction p of edges
Reposition on of their endpoints
Netlogo: Small Worlds
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
Watts-Strogatz model:
Generating small world graphs
• 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
From www.cse.unr.edu/~mgunes/cs765/cs790f10/Lect18_SmallWorld.ppt
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
From www.cse.unr.edu/~mgunes/cs765/cs790f10/Lect18_SmallWorld.ppt
Watts/Strogatz model:
Change in clustering coefficient and average path length as a function of
the proportion of rewired edges
C(p)/C(0)
l(p)/l(0)
1% of links rewired
10% of links rewired
Source: Watts, D.J., Strogatz, S.H.(1998) Collective dynamics of 'small-world' networks. Nature 393:440-442.
From http://www.dmae.upm.esWebpersonalBartolo/
Structured network
• high clustering
• large diameter
• regular
N = 1000 k =10
D = 100 L = 49.51
C = 0.67
Small-world network
• high clustering
• small diameter
• almost regular
N =1000 k = 8-13
D = 14 d = 11.1
C = 0.63
Random network
• small clustering
• small diameter
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)
part of WWW
Typical structure of
Typical structure of
a randomly connected
World Wide Web
network
(nodes = web pages, links =
links between pages)
http://www.dichotomistic.com/images/random
%20network.gif
Concept of “Degree Distribution”
A node with degree 3
Concept of “Degree Distribution”
A node with degree 3
Concept of “Degree Distribution”
A node with degree 3
Number of
Nodes
6
5
4
3
2
1
0
1
2
3
4
5
6
Degree
7
8
9
10
Number of nodes
Number of nodes
part of WWW
Degree
Degree
Number of nodes
Number of nodes
part of WWW
Degree
Degree
Number
nodes
of nodes
Number of
The Web’s approximate Degree Distribution
Degree
Number
nodes
of nodes
Number of
The Web’s approximate Degree Distribution
Degree
Number
nodes
of nodes
Number of
The Web’s approximate Degree Distribution
Degree
Number
nodes
of nodes
Number of
The Web’s approximate Degree Distribution
Degree
Number of nodes
The Web’s approximate Degree Distribution
Degree
The Web’s approximate Degree Distribution
Number of nodes
“Scale-free” distribution
Degree
The Web’s approximate Degree Distribution
Number of nodes
“Scale-free” distribution
Number of nodes with degree k 

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
The Web’s approximate Degree Distribution
Number of nodes
“Scale-free” distribution
Number of nodes with degree k 
“Scale-free” distribution
= “power law” distribution
Degree
“power law”
1
k2
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/hollandcit
ypopulation1.png
The scale-free structure of the Web helps to explain
why Google works so well
part of WWW
The scale-free structure of the Web helps to explain
why Google works so well
part of WWW
It also explains some of the success of other scalefree networks in nature!
Scale-Free Networks are “fractal-like”
http://en.wikipedia.org/wiki/File:WorldWideWebAroundGoogle.png
Scale-Free Networks have high clustering
High
Clustering:
Low Clustering:
part of WWW
High-Clustering Helps in Discovering Community Structure
in Networks
How are Scale-Free Networks Created?
Web pages
Web pages
Web pages
Preferential attachment demo
(Netlogo)
Robustness of Scale-Free Networks
Robustness of Scale-Free Networks
• Vulnerable to targeted “hub” failure
Robustness of Scale-Free Networks
• Vulnerable to targeted “hub” failure
• Robust to random node failure
Robustness of Scale-Free Networks
• Vulnerable to targeted “hub” 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
http://earthobservatory.nasa.gov/
images/imagerecords/3000/3719/
NE_US_OLS2003227.jpg
9:29pm
1 day before
9:14pm
Day of blackout
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_analysi
s_modeling/Random_Normal_Distribution.gif
Normal (“bell-curve) distribution
“Events in ‘tail’ are highly unlikely”
http://resources.esri.com/help/9.3/arcgisdesktop/com/gp_toolref/process_simulations_sensitivity_analysis_and_error_analysi
s_modeling/Random_Normal_Distribution.gif
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
Power law (“scale free”) distribution
http://www.marketoracle.co.uk/images/mauldin_16_10_07image003.gif
Power law (“scale free”) distribution
“More normal than ‘normal’”
http://www.marketoracle.co.uk/images/mauldin_16_10_07image003.gif
“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
Power law (“scale free”) distribution
“More normal than ‘normal’”
http://www.marketoracle.co.uk/images/mauldin_16_10_07image003.gif
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]