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**Classes will begin 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 Theory Applications

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The Course

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THE CLASSES ~1 hour Networks ~20 min Other Topics on Complexity (Bonus Section) NETWORKS Class 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 SECTION Class 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**

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

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EMERGENCE 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 need Something more to Get something more” TED Talk (2007)” **Phillip Anderson “More is Different” Science 177:393–396 (1972) *Adam Smith “The Wealth of Nations” (1776)

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**20 billion neurons 60 trillion synapses**

In addition to the neocortex, the cortical region of the human brain contains more primitive components called the olfactory cortex and the hippocampus that occur in reptiles as well as mammals. The mammalian versions of these structures, however, are associated with other regions of the cortex, hypothalamus, and thalamus in a ring-like assembly centered around the brainstem known as the limbic system. The emotional responses or feelings that mammals experience are produced by the limbic system, which closely interacts with other parts of the brain. The limbic system also is a functional center for long-term memory. 20 billion neurons 60 trillion synapses

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WHY NETWORKS?

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**NETWORKS = ARCHITECTURE OF COMPLEXITY**

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Networks Economics Emergence of Scaling in Random Networks - R Albert, AL Barabási - Science, 1999 Cited by Innovation and Growth in the Global Economy GM Grossman, E Helpman Cited by 4542 Statistical mechanics of complex networks - R Albert, AL Barabási - Reviews of Modern Physics, Cited by 3132 Technical Change, Inequality, and the Labor Market - D Acemoglu - Journal of Economic Literature, Cited by 911 Collective dynamics of'small-world' networks - Find Harvard DJ WATTS, SH STROGATZ - Nature, Cited by 6595 The Market for Lemons: Quality Uncertainty and the Market Mechanism GA Akerlof - Cited by 4561 The structure and function of complex networks - MEJ Newman - Arxiv preprint cond-mat/ , Cited by 2451 The Pricing of Options and Corporate Liabilities F Black, M Scholes - Journal of Political Economy, Cited by 9870

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Networks? 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 Graphs 2 years 4 years 1 year 7 years**

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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**

List of Edges or Links A B B D A C A F B G G F A S 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= 6x x1012 ~5.9x1018

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**A network is a “space”. Networks? 1 2 3 4 5 6 7**

What if we start making neighbors of non-consecutive numbers? Cartesian Space (Lattice) 2-d 1 3 4 5 6 7 2 Now we have different paths between One 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)**

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 Euler's solution to the Königsberg Bridge problem leads to a result that applies to all graphs. Before we see this however, we need some more terminology. Firstly, a Eulerian path is a route from one vertex to another in a graph, using up all the edges in the graph. A Eulerian circuit is a Eulerian path, where the start and end points are the same. This is equivalent to what would be required in the problem. Given these terms a graph is Eulerian if there exists an Eulerian circuit, and Semi-Eulerian if there exists a Eulerian path that is not a circuit. Finally, the degree of a vertex is the number of edges that lead from it. The result above showed that a graph can only be Eulerian if all vertices have an even number of edges from them. In other words, each vertex must have an even degree. It was later proven that any graph with all vertices of even degree will be Eulerian. Similarly if and only if a graph has only 2 vertices with odd order, it will be Semi-Eulerian. Leonhard Euler

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We have taken the data from National Institute on Money in State Politics a nonpartisan organization dedicated to the documentation and research on campaign financing at the state level. For now we have used only data about Governors elected in 2006 and the top 100 donors. Currently we show only donors who donated to more than one candidate or a single donation bigger than $.

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PNAS 2005

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**The Political Blogosphere and the 2004 U. S**

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

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RANDOM GRAPH THEORY

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**Random Graph Theory Erdos-Renyi Model (1959) Paul Erdos**

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. Alfred Renyi

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**Random Graph Theory: Erdos-Renyi (1959)**

Subgraphs Trees Cycles k Cliques k k(k-1)/2 Nodes: Links: k k-1

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**GN,p F(k,l) CNk Nk pl /a k! a pl E=**

Random Graph Theory: Erdos-Renyi (1959), Bollobas (1985) CNk Can choose the k nodes in N choose k ways GN,p Nk pl /a Which in the large N goes like F(k,l) We can permute the nodes we choose in k! ways, but have to remember not to double count isomorphisms (a) k! a pl Each link occurs with Probability p E=

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**E Nk pl /a p(N)~ cN-k/l E=cl/a=l**

Random Graph Theory: Erdos-Renyi (1959), Bollobas (1985) E Nk pl /a In the threshold: p(N)~ cN-k/l Which implies a number of subgraphs: E=cl/a=l Bollobas (1985) R. Albert, A.-L. Barabasi, Rev. Mod. Phys (2002)

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**Subgraphs appear suddenly (percolation threshold)**

Probability of having a property Question for the class: Given that the critical connectivity is p(N)~ cN-k/l When does a random graph become connected? p

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**Random Graph Theory: Erdos-Renyi (1959)**

Degree Distribution K=8 Binomial distribution For large N approaches a poison distribution K=4

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**Random Graph Theory: Erdos-Renyi (1959) Clustering**

Ci=triangles/possible triangles Clustering Coefficient = <C>

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**Distance Between A and B?**

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**Random Graph Theory: Erdos-Renyi (1959) Average Path Length**

Number of nodes at distance m from a randomly chosen node Hence the average path length is <k>4 <k>3 <k>2 <k> m

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IT IS A SMALL WORLD

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**Six Degrees (Stanley Milgram)**

1 person 160 people Stanley Milgram

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**But only a few chains were ever completed!**

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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**R. Albert, A-L Barabasi, Rev. Mod. Phys. 2002**

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L Steps needed for completion Attrition rates

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Median L=7 Same Country Median L = 5 Cross Country Medial L = 7

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Total number of linkable actors: Weighted total of linkable actors: Average Kevin Bacon number: 2.946 Kevin Bacon Number # of People 1 2108 2 204188 3 601747 4 136178 5 8656 6 839 7 111 8 12 Kevin Bacon

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**Average Connery number: 2.731**

# of people 1 2272 2 218560 3 380721 4 40263 5 3537 6 535 7 66 8 Sean Connery

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**Hollywood Revolves Around**

Click on a name to see that person's table. Steiger, Rod ( ) Lee, Christopher (I) ( ) Hopper, Dennis ( ) Sutherland, Donald (I) ( ) Keitel, Harvey ( ) Pleasence, Donald ( ) von Sydow, Max ( ) Caine, Michael (I) ( ) Sheen, Martin ( ) Quinn, Anthony ( ) Heston, Charlton ( ) Hackman, Gene ( ) Connery, Sean ( ) Stanton, Harry Dean ( ) Welles, Orson ( ) Mitchum, Robert ( ) Gould, Elliott ( ) Plummer, Christopher (I) ( ) Coburn, James ( ) Borgnine, Ernest ( ) Rod Steiger

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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): XXXXXX Number # of people 1 2 7 3 4 21 5 28 6 15 115 8 44700 9 440047 10 148900 11 10764 12 1158 13 183 14

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**Kevin Bacon has +2000 co-workers, so does Sean Connery, while the worst connected**

actor in Hollywood has just 1. Are networks random? TO BE CONTINUED…….

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**Determinism ≠ Predictability**

BONUS SECTION: CHAOS Determinism ≠ Predictability

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**Lorenz Attractor Edward Lorenz**

Lorenz, E. N. (1963). "Deterministic nonperiodic flow". J. Atmos. Sci. 20: 130–141 Lorenz Attractor Edward Lorenz

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**By m = 2 there are limit cycles of every possible length!!!**

Xn+1=Xn The Tent Map Xn+1=m(1-2|xn-1/2|) By m = 2 there are limit cycles of every possible length!!!

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**http://www. geom. uiuc. edu/~math5337/ds/applets/burbanks/Logistic**

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Xn+1=Xn+9Xn The circle hiker Origin X1 Xn+1=significand(10Xn) 9x0 X0 X0= X1= X2= X3= ….

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**Remember Not to Always Blame the Butterfly**

David Orrell

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**Take Home Messages NETWORKS**

-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 size L~ 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.

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