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Classes will begin shortly

Classes will begin shortly. Networks, Complexity and Economic Development. Characterizing the Structure of Networks Cesar A. Hidalgo PhD. WWW. Expected. P( k ) ~ k - . Found. Over 3 billion documents. Exponential Network. Scale-free Network.

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  1. Classes will begin shortly

  2. Networks, Complexity and Economic Development Characterizing the Structure of Networks Cesar A. Hidalgo PhD

  3. WWW Expected P(k) ~ k- Found Over 3 billion documents Exponential Network Scale-free Network R. Albert, H. Jeong, A-L Barabasi, Nature, 401 130 (1999).

  4. Take home messages -Networks might look messy, but are not random. -Many networks in nature are Scale-Free (SF), meaning that just a few nodes have a disproportionately large number of connections. -Power-law distributions are ubiquitous in nature. -While power-laws are associated with critical points in nature, systems can self-organize to this critical state. - There are important dynamical implications of the Scale-Free topology. -SF Networks are more robust to failures, yet are more vulnerable to targeted attacks. -SF Networks have a vanishing epidemic threshold.

  5. Local Measures

  6. CENTRALITY MEASURES Measure the “importance” of a node in a network.

  7. Hollywood Revolves Around 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) Rod Steiger

  8. Most Connected Actors in Hollywood (measured in the late 90’s) Mel Blanc 759 Tom Byron 679 Marc Wallice 535 Ron Jeremy 500 Peter North 491 TT Boy 449 Tom London 436 Randy West 425 Mike Horner 418 Joey Silvera 410 XXX A-L Barabasi, “Linked”, 2002

  9. DEGREE CENTRALITY K= number of links Where Aij = 1 if nodes i and j are connected and 0 otherwise

  10. BETWENNESS CENTRALITY BC(G)=0 BC(D)=9+1/2=9.5 BC= number of shortest Paths that go through a node. BC(B)=4*6=24 BC(A)=5*5+4=29 G I D A J B H C E K F N=11

  11. C(G)=1/10(1+2*3+2*3+4+3*5) C(G)=3.2 C(A)=1/10(4+2*3+3*3) C(A)=1.9 CLOSENESS CENTRALITY C= Average Distance to neighbors G I D A J B H C E N=11 K F C(B)=1/10(2+2*6+2*3) C(B)=2

  12. EIGENVECTOR CENTRALITY Consider the Adjacency Matrix Aij = 1 if node i is connected to node jand 0 otherwise. Consider the eigenvalue problem: Ax=lx Then the eigenvector centrality of a node is defined as: where l is the largest eigenvalue associated with A.

  13. PAGE RANK PR=Probability that a random walker with interspersed Jumps would visit that node. PR=Each page votes forits neighbors. K G B A H E F J C I D PR(A)=PR(B)/4 + PR(C)/3 + PR(D)+PR(E)/2 A random surfer eventually stops clicking PR(X)=(1-d)/N + d(SPR(y)/k(y))

  14. PAGE RANK PR=Probability that a random Walker would visit that node. PR=Each page votes forits neighbors.

  15. CLUSTERING MEASURES Measure the density of a group of nodes in a Network

  16. Clustering Coefficient Ci=2D/k(k-1) CA=2/12=1/6 CC=2/2=1 CE=4/6=2/3 G I D A J B H C E K F

  17. Topological Overlap Mutual Clustering TO(A,B)=Overlap(A,B)/NormalizingFactor(A,B) TO(A,B)=N(A,B)/max(k(A),k(B)) TO(A,B)=N(A,B)/min(k(A),k(B)) TO(A,B)=N(A,B)/ (k(A)xk(B))1/2 TO(A,B)=N(A,B)/(k(A)+k(B)) G I D A J B H C E K F

  18. Topological Overlap Mutual Clustering TO(A,B)=N(A,B)/max(k(A),k(B)) TO(A,B)=0 TO(A,D)=1/4 TO(E,D)=2/4 G I D A J B H C E K F

  19. MOTIFS

  20. Motifs

  21. Structural Equivalence

  22. Global Measures

  23. The Distribution of any of the previously introduced measures

  24. Giant Component Components S=NumberOfNodesInGiantComponent/TotalNumberOfNodes

  25. Diameter

  26. Diameter=Maximum Distance Between Elements in a Set Diameter=D(G,J)=D(C,J)=D(G,I)=…=5 G I D A J B H C E K F

  27. Average Path Length A B C D E F G A B C D E F G 1 2 1 1 1 2 3 2 2 2 3 1 1 3 2 1 2 1 2 2 3 G D A B C D(1)=8 D(2)=9 D(3)=4 E F L=(8+2x9+3x4)/(8+9+4) L=1.8

  28. Degree CorrelationsAre Hubs Connected to Hubs?

  29. Phys. Rev. Lett. 87, 258701 (2001)

  30. Wait:are we comparing to the right thing

  31. Compared to what? Randomized Network J J K K I I H H G I D B B A J B H C A A F F E K F E E D D G C G C

  32. Physica A 333, 529-540 (2004) Randomized Network Internet

  33. Connectivity Pattern/Randomized Z-score Connectivity Pattern/Randomized

  34. Data Models

  35. After Controlling for Randomized Network Data Models

  36. Fractal Networks

  37. Mandelbrot BB

  38. Generating Koch Curve Measuring the Dimension of Koch Curve

  39. White Noise Pink Noise Brown Noise

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