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Classes will Classes will begin shortlybegin shortly
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Networks, Complexity and Economic Development
Class 1: Random Graphs and Small World Networks
Cesar A. Hidalgo PhD
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Please allow me to introduce myself..
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The Course
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1 2 3 4 5 6 7
Theory Applications
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The Course
1 2 3 4 5 6 7
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THE CLASSES
1 hour Networks~20 min Other Topics on Complexity (Bonus Section)
NETWORKSClass 1: Random networks, simple graphs and basic network characteristics.Class 2: Scale-Free Networks.Class 3: Characterizing Network Topology.Class 4: Community Structure.Class 5: Network Dynamics.Class 6: Networks in Biology.Class 7: Networks in Economy.
BONUS SECTIONClass 1: Chaos.Class 2: Fractals. Power-Laws. Self-Organized Criticality. Class 3: Drawing your own Networks using Cytoscape.Class 4: Community finding software.Class 5: Crowd-sourcing.Class 6: Synthetic Biology.Class 7: TBA.
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COMPLEX SYSTEMS
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Complex Systems:
-Large number of parts-Properties of parts are heterogeneously distributed-Parts interact through a host of non-trivial interactions
Components:
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**Phillip Anderson“More is Different”Science 177:393–396(1972)
*Adam Smith“The Wealth of Nations”(1776)
An aggregate system is not equivalent to the sum of its parts.
People’s action can contribute to ends which are no part of their intentions. (Smith)*
Local rules can produce emergent global behavior
For example: The global match between supply and demand
More is different (Anderson)**
There is emerging behavior in systems that escape local explanations. (Anderson)
**Murray Gell-Mann“You do not needSomething more to Get something more”TED Talk (2007)”
EMERGENCE
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20 billion neurons60 trillion synapses
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WHY NETWORKS?
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NETWORKS = ARCHITECTURE OF COMPLEXITY
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Emergence of Scaling in Random Networks - R Albert, AL Barabási - Science, 1999Cited by 3872 -
Statistical mechanics of complex networks - R Albert, AL Barabási - Reviews of Modern Physics, 2002 Cited by 3132
Collective dynamics of'small-world' networks - Find It @ HarvardDJ WATTS, SH STROGATZ - Nature, 1998 Cited by 6595
The structure and function of complex networks - MEJ Newman - Arxiv preprint cond-mat/0303516, 2003 - Cited by 2451
Innovation and Growth in the Global Economy
GM Grossman, E Helpman - 1991 - Cited by 4542 Technical Change, Inequality, and the Labor M
arket -
D Acemoglu - Journal of Economic Literature, 2002 -
Cited by 911The Market for Lemons: Quality Uncertainty
and the Market Mechanism -1970- GA Akerlof -
Cited by 4561
The Pricing of Options and Corporate Liabilities
F Black, M Scholes - Journal of Political Economy, 1973
Cited by 9870
Networks Economics
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Networks?
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We all had some academic experience with networks at some point in our lives
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Types of Networks
• Simple Graph. Symmetric, Binary.Example: Countries that share a border in South America
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Types of Networks
• Bi-Partite Graph
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Types of Networks
• Directed Graphs
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Types of Networks
• Weighted Graphs4 years
7 years
2 years
1 year
1 year 3 years (1 / 2)
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Simple Graph:
Symmetric, Binary.
Directed Graph:
Non-Symmetric, Binary.
Directed and Weighted Graph:
Any Matrix
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Networks are usually sparser than matrices
A BB DA CA FB GG FA S
List of Edges or Links
A B
D
c
F
G
S
Example: The World Social Network
Nodes = 6x109
Links=103 x 6x109/2 = 3x1012
Possible Links= (6x109-1)x 6x109/2 = 6x1018
Number of Zeros= 6x1018 - 3x1012 5.9x1018
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Networks?
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A network is a “space”.
Cartesian Space (Lattice)2-d
1 2 3 4 5 6 7
What if we start making neighborsof non-consecutive numbers?
1
3 4 5
672
Now we have different paths betweenOne number and another
Cartesian Space (Lattice)1-d
1 2 3 4 5 6 7
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Networks now and then
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Konigsberg bridge problem, Euler (1736)
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Leonhard Euler
Eulerian path: is a route from one vertex to another in a graph, using up all the edges in the graph
Eulerian circuit: is a Eulerian path, where the start and end points are the same
A graph can only be Eulerian if all vertices have an even number of edges
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28http://www.weshowthemoney.com/
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29PNAS 2005
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30The Political Blogosphere and the 2004 U.S. Election: Divided They BlogLada A. Adamic and Natalie Glance, LinkKDD-2005
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31http://presidentialwatch08.com/index.php/map/
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33http://www.blogopole.fr/
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http://prezoilmoney.oilchangeusa.org/
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RANDOM GRAPH THEORY
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Erdos-Renyi Model (1959)
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Original Formulation:N nodes, n links chosen randomly from the N(N-1)/2 possible links.
Alternative Formulation:N nodes. Each pair is connected with probability p.Average number of links =p(N(N-1))/2;
Random Graph Theory Works on the limit N-> and studies when do properties on a graph emerge as function of p.
Random Graph Theory
Paul Erdos
Alfred Renyi
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Random Graph Theory: Erdos-Renyi (1959)
Subgraphs
Trees
Nodes:Links:
kk-1
Cycles
kk
Cliques
kk(k-1)/2
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Random Graph Theory: Erdos-Renyi (1959), Bollobas (1985)
GN,p
F(k,l) CNk
Can choose the k nodes in N choose
k ways
pl
Each link occurs withProbability p
We can permute the nodes we choosein k! ways, but have to remember not to double
count isomorphisms (a)
k!a
Nk pl /a
Which in the large Ngoes like
E=
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E Nk pl /a
In the threshold:
Random Graph Theory: Erdos-Renyi (1959), Bollobas (1985)
p(N)cN-k/l
Which implies a number of subgraphs:
E=cl/a=
Bollobas (1985)
R. Albert, A.-L. Barabasi, Rev. Mod. Phys (2002)
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41p
Prob
abili
ty o
f hav
ing
a pr
oper
ty
Subgraphs appear suddenly (percolation threshold)
Question for the class:
Given that the criticalconnectivity is p(N)cN-k/l
When does a random graphbecome connected?
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Random Graph Theory: Erdos-Renyi (1959) Degree Distribution
K=8
K=4
Binomial distribution
For large N approaches a poison distribution
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Random Graph Theory: Erdos-Renyi (1959) Clustering
Ci=triangles/possible trianglesClustering Coefficient = <C>
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A
B
Distance Between A and B?
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Random Graph Theory: Erdos-Renyi (1959) Average Path Length
Number of nodes at distancem from a randomly chosen node
Hence the average path length is
m
<k>
<k>2
<k>3
<k>4
lkN
kN l
~)log(/)log(
~
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IT IS A SMALL WORLD
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Six Degrees (Stanley Milgram)
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Stanley Milgram
160 people
1 person
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Stanley Milgram found that the average length of the chain connecting the sender and receiver was of length 5.5.
But only a few chains were ever completed!
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Duncan Watts
Steve Strogatz
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53R. Albert, A-L Barabasi, Rev. Mod. Phys. 2002
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Attrition rates
L
Steps needed for completion
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Median L=7Same Country Median L = 5Cross Country Medial L = 7
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Kevin Bacon Number # of People
0 1
1 2108
2 204188
3 601747
4 136178
5 8656
6 839
7 111
8 12
Total number of linkable actors: 953840Weighted total of linkable actors: 2809624Average Kevin Bacon number: 2.946
Kevin Bacon
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Connery Number # of people
0 1
1 2272
2 218560
3 380721
4 40263
5 3537
6 535
7 66
8 2
Average Connery number: 2.731
Sean Connery
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Rod Steiger
Click on a name to see that person's table. Steiger, Rod (2.678695) Lee, Christopher (I) (2.684104) Hopper, Dennis (2.698471) Sutherland, Donald (I) (2.701850) Keitel, Harvey (2.705573) Pleasence, Donald (2.707490) von Sydow, Max (2.708420) Caine, Michael (I) (2.720621) Sheen, Martin (2.721361) Quinn, Anthony (2.722720) Heston, Charlton (2.722904) Hackman, Gene (2.725215) Connery, Sean (2.730801) Stanton, Harry Dean (2.737575) Welles, Orson (2.744593) Mitchum, Robert (2.745206) Gould, Elliott (2.746082) Plummer, Christopher (I) (2.746427) Coburn, James (2.746822) Borgnine, Ernest (2.747229)
Hollywood Revolves Around
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XXXXXX Number # of people
0 1
1 1
2 7
3 2
4 21
5 28
6 15
7 115
8 44700
9 440047
10 148900
11 10764
12 1158
13 183
14 14
15 1
Is there a "worst" center (or most obscure actor) in the Hollywood universe?Of course. I won't tell you the name of the person who produces the highest
average number in the IMDb, but his/her table looks like this (as of June 29, 2004):
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Kevin Bacon has +2000 co-workers, so does Sean Connery, while the worst connectedactor in Hollywood has just 1.
Are networks random?
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BONUS SECTION:CHAOS
Determinism ≠ Predictability
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64Edward Lorenz
Lorenz AttractorLorenz, E. N. (1963). "Deterministic nonperiodic flow". J. Atmos. Sci. 20: 130–141
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The Tent MapXn+1=XnXn+1=(1-2|xn-1/2|)
By = 2 there are limit cycles
of every possible length!!!
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http://www.geom.uiuc.edu/~math5337/ds/applets/burbanks/Logistic.html
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X0
Xn+1=Xn+9Xn
The circle hiker Origin
9x0
X1Xn+1=significand(10Xn)
X0=0.314159..X1=0.141592.X2=0.415926..X3=0.159265..
….
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Remember Not to Always Blame the Butterfly
David Orrell
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Take Home MessagesNETWORKS-Networks can be used to represent a wide set of systems-The properties of random networks emerge suddenly as a function of connectivity.-The distance between nodes in random networks is small compared to network sizeL log(N)-Networks can exhibit simultaneously: short average path length and high clustering(SMALL WORLD PROPERTY)-The coexistence of these last two properties cannot be explained by random networks-The small world property of networks is not exclusive of “social” networks.
BONUS-Deterministic Systems are not necessarily predictable.-But you shouldn’t always blame the butterfly.