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

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

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.

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XXX Most Connected Actors in Hollywood (measured in the late 90s) A-L Barabasi, Linked, 2002 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

DEGREE CENTRALITY K= number of links Where A ij = 1 if nodes i and j are connected and 0 otherwise

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

CLOSENESS CENTRALITY C= Average Distance to neighbors A BH I J K D G E C F N=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 C(B)=1/10(2+2*6+2*3) C(B)=2

EIGENVECTOR CENTRALITY Consider the Adjacency Matrix A ij = 1 if node i is connected to node j and 0 otherwise. Consider the eigenvalue problem: Ax= x Then the eigenvector centrality of a node is defined as: where 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. A EF G H I B K C J 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( PR(y)/k(y))

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 A BH I J K D G E C F C i =2 /k(k-1) C A =2/12=1/6 C C =2/2=1C E =4/6=2/3

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

Topological Overlap Mutual Clustering A BH I J K D G E C F 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

MOTIFS

Motifs

Structural Equivalence

Global Measures

The Distribution of any of the previously introduced measures

Giant Component Components S=NumberOfNodesInGiantComponent/TotalNumberOfNodes

Diameter

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

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

Degree Correlations Are Hubs Connected to Hubs?

Phys. Rev. Lett. 87, (2001)

Wait: are we comparing to the right thing

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

Internet Randomized Network Physica A 333, (2004)

Connectivity Pattern/RandomizedZ-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 Fractal Network Non 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