network thinking - computer action teamweb.cecs.pdx.edu/.../networks.pdfthe network has relatively...
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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?
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?
The Science 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
Barack Obama
my mother
Nancy Bekavac
Small-World Property (Watts and Strogatz, 1998)
me
Barack Obama
my mother
Nancy Bekavac
Hillary Clinton
Small-World Property (Watts and Strogatz, 1998)
me
Barack Obama
my mother
Nancy Bekavac
Hillary Clinton
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
Barack Obama
Patrick Leahy
my cousin Matt Dunne
Small-World Property (Watts and Strogatz, 1998)
me
Barack Obama
Patrick Leahy
my cousin Matt Dunne
Stanley Milgram
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”
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”.
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?
The Small-World Property
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.
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
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)
From http://www.dmae.upm.esWebpersonalBartolo/
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
From http://www.dmae.upm.esWebpersonalBartolo/
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
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
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.
From www.cse.unr.edu/~mgunes/cs765/cs790f10/Lect18_SmallWorld.ppt
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
From http://www.dmae.upm.esWebpersonalBartolo/
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”
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
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
The Web’s approximate Degree Distribution N
umbe
r of n
odes
Degree
Num
ber o
f nod
es
The Web’s approximate Degree Distribution N
umbe
r of n
odes
Degree
Degree
The Web’s approximate Degree Distribution N
umbe
r of n
odes
Degree
“Scale-free” distribution
The Web’s approximate Degree Distribution N
umbe
r of n
odes
Degree
“Scale-free” distribution
€
Number of nodes with degree k ≈ 1k 2
The Web’s approximate Degree Distribution N
umbe
r of n
odes
Degree
“Scale-free” distribution
The Web’s approximate Degree Distribution N
umbe
r of n
odes
“power law”
Degree
“Scale-free” distribution
The Web’s approximate Degree Distribution N
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
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
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
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
“More normal than ‘normal’ ”
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
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]