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

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.

COMPLEX SYSTEMS

Complex Systems:

-Large number of parts-Properties of parts are heterogeneously distributed-Parts interact through a host of non-trivial interactions

Components:

**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

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

Networks?

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We all had some academic experience with networks at some point in our lives

Types of Networks

• Simple Graph. Symmetric, Binary.Example: Countries that share a border in South America

Types of Networks

• Bi-Partite Graph

Types of Networks

• Directed Graphs

Types of Networks

• Weighted Graphs4 years

7 years

2 years

1 year

1 year 3 years (1 / 2)

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

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

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

28http://www.weshowthemoney.com/

29PNAS 2005

30The Political Blogosphere and the 2004 U.S. Election: Divided They BlogLada A. Adamic and Natalie Glance, LinkKDD-2005

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

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=

40

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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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

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

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.