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Networks, Complexity and Economic Development Characterizing the Structure of Networks Cesar A. Hidalgo PhD Exponential Network Over 3 billion documents Scale-free Network P(k) ~ k- R. Albert, H. Jeong, A-L Barabasi, Nature, 401 130 (1999). 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. Local Measures CENTRALITY MEASURES Measure the “importance” of a node in a network. 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 DEGREE CENTRALITY K= number of links Where Aij = 1 if nodes i and j are connected and 0 otherwise BETWENNESS CENTRALITY BC(G)=0 BC(D)=9+1/2=9.5 BC(A)=5*5+4=29 BC= number of shortest Paths that go through a node. BC(B)=4*6=24 G I D A C E F B J H K N=11 C(G)=1/10(1+2*3+2*3+4+3*5) C(G)=3.2 CLOSENESS CENTRALITY C(A)=1/10(4+2*3+3*3) C(A)=1.9 C= Average Distance to neighbors G I D A C B E J H K F C(B)=1/10(2+2*6+2*3) C(B)=2 N=11 EIGENVECTOR CENTRALITY Consider the Adjacency Matrix Aij = 1 if node i is connected to node j and 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. PAGE RANK PR=Probability that a random walker with interspersed Jumps would visit that node. PR=Each page votes for its neighbors. K G B A J E C 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)) H F I PAGE RANK PR=Probability that a random Walker would visit that node. PR=Each page votes for its neighbors. CLUSTERING MEASURES Measure the density of a group of nodes in a Network 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 C B J H E F K 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 C B J H E F K 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 C B J H E F K MOTIFS Motifs Structural Equivalence Global Measures The Distribution of any of the previously introduced measures Components Giant Component S=NumberOfNodesInGiantComponent/TotalNumberOfNodes Diameter Diameter=Maximum Distance Between Elements in a Set Diameter=D(G,J)=D(C,J)=D(G,I)=…=5 G I D A C E F B J H K Average Path Length A B C D E F G A B C D E F G 1 2 1 3 2 1 1 2 1 1 1 2 3 2 2 2 3 2 1 2 3 D(1)=8 D(2)=9 D(3)=4 L=(8+2x9+3x4)/(8+9+4) L=1.8 G D A C E F B Degree Correlations Are Hubs Connected to Hubs? Phys. Rev. Lett. 87, 258701 (2001) Wait: are we comparing to the right thing Compared to what? J F E C D G E F A B H I K K J B A C G D F E C D G H I H A B I K J Randomized Network Physica A 333, 529-540 (2004) Randomized Network Internet Connectivity Pattern/Randomized Z-score Connectivity Pattern/Randomized Data Models After Controlling for Randomized Network Data Models Fractal Networks Mandelbrot BB Generating Koch Curve Measuring the Dimension of Koch Curve White Noise Pink Noise Brown Noise Attack Tolerance Non Fractal Network Fractal Network Take Home Messages -To characterize the structure of a Network we need many different measures -This measures allow us to differentiate between the different networks in nature Today we saw: Local Measures Centrality measures (degree, closeness, betweenness, eigenvector, page-rank) Clustering measures (Clustering, Topological Overlap or Mutual Clustering) Motifs Global Measures Degree Correlations, Correlation Profile. Hierarchical Structure Fractal Structure Connections Between Local and Global Measures