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Classes will begin shortly 1

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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.. 3

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The Course Theory Applications

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5 The Course

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6 THE CLASSES 1 hour Networks ~20 minOther 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 -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. Peoples 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) EMERGENCE

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20 billion neurons 60 trillion synapses

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13 WHY NETWORKS?

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15 NETWORKS = ARCHITECTURE OF COMPLEXITY

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17 Emergence of Scaling in Random NetworksEmergence of Scaling in Random Networks - R Albert, AL Barabási - Science, 1999 Cited by 3872Cited by Statistical mechanics of complex networksStatistical mechanics of complex networks - R Albert, AL Barabási - Reviews of Modern Physics, 2002 Cited by 3132 Cited by 3132 Collective dynamics of'small-world' networksCollective dynamics of'small-world' networks - Find Harvard DJ WATTS, SH STROGATZ - Nature, 1998 Cited by 6595 Find Harvard Cited by 6595 The structure and function of complex networksThe structure and function of complex networks - MEJ Newman - Arxiv preprint cond- mat/ , Cited by 2451 Cited by 2451 Innovation and Growth in the Global Economy GM Grossman, E Helpman Cited by 4542Innovation and Growth in the Global Economy Cited by 4542 Technical Change, Inequality, and the Labor MarketTechnical Change, Inequality, and the Labor Market - D Acemoglu - Journal of Economic Literature, Cited by 911 Cited by 911 The Market for Lemons: Quality Uncertainty and the Market Mechanism GA Akerlof - Cited by 4561 Cited by 4561 The Pricing of Options and Corporate Liabilities The Pricing of Options and Corporate Liabilities F Black, M Scholes - Journal of Political Economy, 1973 Cited by 9870 Cited by 9870 NetworksEconomics

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Networks? 18 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 4 years 7 years 2 years 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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24 Networks are usually sparser than matrices ABBDACAFBGGFASABBDACAFBGGFAS List of Edges or Links A B D c F G S Example: The World Social Network Nodes = 6x10 9 Links=10 3 x 6x10 9 /2 = 3x10 12 Possible Links= (6x )x 6x10 9 /2 = 6x10 18 Number of Zeros= 6x x x10 18

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Networks? 25 A network is a space. Cartesian Space (Lattice) 2-d What if we start making neighbors of non-consecutive numbers? Now we have different paths between One number and another Cartesian Space (Lattice) 1-d

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26 Networks now and then

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Konigsberg bridge problem, Euler (1736) 27 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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29 PNAS 2005

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

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36 RANDOM GRAPH THEORY

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Erdos-Renyi Model (1959) 37 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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38 Random Graph Theory: Erdos-Renyi (1959) Subgraphs Trees Nodes: Links: k k-1 Cycles kkkk Cliques k k(k-1)/2

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

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

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

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43 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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44 Random Graph Theory: Erdos-Renyi (1959)Clustering C i =triangles/possible triangles Clustering Coefficient =

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45 A B Distance Between A and B?

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46 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 m 2 3 4

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47 IT IS A SMALL WORLD

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Six Degrees (Stanley Milgram) 48 Stanley Milgram 160 people 1 person

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49 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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51 Duncan Watts Steve Strogatz

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

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55 Attrition rates L Steps needed for completion

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

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58 Kevin BaconKevin Bacon Number# of People Total number of linkable actors: Weighted total of linkable actors: Average Kevin Bacon number: Kevin Bacon

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59 Connery Number# of people Average Connery number: Sean Connery

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60 Rod Steiger Click on a name to see that person's table. Steiger, RodSteiger, Rod ( ) Lee, Christopher (I)Lee, Christopher (I) ( ) Hopper, DennisHopper, Dennis ( ) Sutherland, Donald (I)Sutherland, Donald (I) ( ) Keitel, HarveyKeitel, Harvey ( ) Pleasence, DonaldPleasence, Donald ( ) von Sydow, Maxvon Sydow, Max ( ) Caine, Michael (I)Caine, Michael (I) ( ) Sheen, MartinSheen, Martin ( ) Quinn, AnthonyQuinn, Anthony ( ) Heston, CharltonHeston, Charlton ( ) Hackman, GeneHackman, Gene ( ) Connery, SeanConnery, Sean ( ) Stanton, Harry DeanStanton, Harry Dean ( ) Welles, OrsonWelles, Orson ( ) Mitchum, RobertMitchum, Robert ( ) Gould, ElliottGould, Elliott ( ) Plummer, Christopher (I)Plummer, Christopher (I) ( ) Coburn, JamesCoburn, James ( ) Borgnine, ErnestBorgnine, Ernest ( ) Hollywood Revolves Around

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61 XXXXXX Number# of people 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):IMDb

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62 Kevin Bacon has co-workers, so does Sean Connery, while the worst connected actor in Hollywood has just 1. Are networks random?

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63 BONUS SECTION: CHAOS Determinism Predictability

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

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65 The Tent Map Xn+1=Xn X n+1 = (1-2|x n -1/2|) By = 2 there are limit cycles of every possible length!!!

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67 X0X0 X n+1 =X n +9X n The circle hikerOrigin 9x 0 X1X1 X n+1 =significand(10X n ) X 0 = X 1 = X 2 = X 3 = ….

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68 Remember Not to Always Blame the Butterfly David Orrell

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69 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 shouldnt always blame the butterfly.

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