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Classes will begin shortly

Classes will begin shortly. Networks, Complexity and Economic Development. Class 1: Random Graphs and Small World Networks Cesar A. Hidalgo PhD. Please allow me to introduce myself. The Course. 1 2 3 4 5 6 7. Theory. Applications. The Course. 1 2 3 4 5 6 7. THE CLASSES.

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Classes will begin shortly

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  1. Classes will begin shortly

  2. Networks, Complexity and Economic Development Class 1: Random Graphs and Small World Networks Cesar A. Hidalgo PhD

  3. Please allow me to introduce myself..

  4. The Course 1 2 3 4 5 6 7 Theory Applications

  5. The Course 1 2 3 4 5 6 7

  6. 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.

  7. COMPLEX SYSTEMS

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

  9. 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)

  10. 20 billion neurons 60 trillion synapses

  11. WHY NETWORKS?

  12. NETWORKS = ARCHITECTURE OF COMPLEXITY

  13. Networks Economics Emergence of Scaling in Random Networks - R Albert, AL Barabási - Science, 1999 Cited by 3872 - Innovation and Growth in the Global EconomyGM Grossman, E Helpman - 1991 - Cited by 4542 Technical Change, Inequality, and the Labor Market - D Acemoglu - Journal of Economic Literature, 2002 - Cited by 911 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 Market for Lemons: Quality Uncertainty and the Market Mechanism -1970- GA Akerlof - Cited by 4561 The structure and function of complex networks -  MEJ Newman - Arxiv preprint cond-mat/0303516, 2003 - Cited by 2451 The Pricing of Options and Corporate LiabilitiesF Black, M Scholes - Journal of Political Economy, 1973 Cited by 9870

  14. Networks? We all had some academic experience with networks at some point in our lives

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

  16. Types of Networks • Bi-Partite Graph

  17. Types of Networks • Directed Graphs

  18. Types of Networks • Weighted Graphs 2 years 4 years 1 year 7 years 1 year 3 years (1 / 2)

  19. Simple Graph: Symmetric, Binary. Directed Graph: Non-Symmetric, Binary. Directed and Weighted Graph: Any Matrix

  20. 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 F G c A B Example: The World Social Network Nodes = 6x109 Links=103 x 6x109/2 = 3x1012Possible Links= (6x109-1)x 6x109/2 = 6x1018 Number of Zeros= 6x1018 - 3x1012~5.9x1018 S D

  21. Networks? A network is a “space”. What if we start making neighbors of non-consecutive numbers? Cartesian Space (Lattice)2-d 1 2 3 4 5 6 7 Now we have different paths between One number and another 2 6 7 Cartesian Space (Lattice) 1-d 1 5 4 3 1 2 3 4 5 6 7

  22. Networks now and then

  23. 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 Leonhard Euler

  24. http://www.weshowthemoney.com/

  25. PNAS 2005

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

  27. http://presidentialwatch08.com/index.php/map/

  28. http://www.blogopole.fr/

  29. http://prezoilmoney.oilchangeusa.org/

  30. RANDOM GRAPH THEORY

  31. Paul Erdos Erdos-Renyi Model (1959) Random Graph Theory 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

  32. Random Graph Theory: Erdos-Renyi (1959) Subgraphs Cycles Cliques Trees k k(k-1)/2 k k Nodes: Links: k k-1

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

  34. 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)

  35. Subgraphs appear suddenly (percolation threshold) Probability of having a property Question for the class: Given that the criticalconnectivity is p(N)~ cN-k/l When does a random graphbecome connected? p

  36. Random Graph Theory: Erdos-Renyi (1959) Degree Distribution K=8 Binomial distribution For large N approaches a poison distribution K=4

  37. Random Graph Theory: Erdos-Renyi (1959) Clustering Ci=triangles/possible triangles Clustering Coefficient = <C>

  38. Distance Between A and B? B A

  39. 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

  40. IT IS A SMALL WORLD

  41. Six Degrees (Stanley Milgram) 1 person 160 people Stanley Milgram

  42. 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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