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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, 401 130 (1999).

Expected

P(k) ~ k-

FoundSca

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ree N

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ork

Exponenti

al N

etw

ork

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.

8

Rod Steiger

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)

Hollywood Revolves Around

XXX

Most Connected Actors in Hollywood(measured in the late 90’s)

A-L Barabasi, “Linked”, 2002

Mel Blanc 759Tom Byron 679

Marc Wallice 535Ron Jeremy 500Peter North 491

TT Boy 449Tom London 436Randy West 425

Mike Horner 418Joey Silvera 410

DEGREE CENTRALITY

K= number of links

Where Aij = 1 if nodes i and j are connected and 0 otherwise

BETWENNESS CENTRALITY

BC= number of shortestPaths that go through a

node.

A B H

I

J

K

DG

E

C

F

BC(G)=0

N=11

BC(D)=9+1/2=9.5

BC(B)=4*6=24

BC(A)=5*5+4=29

CLOSENESS CENTRALITY

C= Average Distanceto neighbors

A B H

I

J

K

DG

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 Aij = 1 if node i is connected to node jand 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 randomwalker with interspersedJumps would visit that node.PR=Each page votes forits neighbors.

A E F

G

H

I

BK

C

J

D

PR(A)=PR(B)/4 + PR(C)/3 + PR(D)+PR(E)/2A random surfer eventually stops clicking

PR(X)=(1-d)/N + d(PR(y)/k(y))

PAGE RANK

PR=Probability that a randomWalker would visit that node.PR=Each page votes forits neighbors.

CLUSTERING MEASURES

Measure the densityof a group of nodes in a

Network

Clustering Coefficient

A B H

I

J

K

DG

E

C

F

Ci=2/k(k-1)

CA=2/12=1/6 CC=2/2=1 CE=4/6=2/3

Topological OverlapMutual Clustering

A B H

I

J

K

DG

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 OverlapMutual Clustering

A B H

I

J

K

DG

E

C

F

TO(A,B)=N(A,B)/max(k(A),k(B))

TO(A,B)=0TO(A,D)=1/4TO(E,D)=2/4

MOTIFS

Motifs

Structural Equivalence

Global Measures

The Distribution of any of the previously introduced measures

Giant ComponentComponents

S=NumberOfNodesInGiantComponent/TotalNumberOfNodes

Diameter

A B H

I

J

K

DG

E

C

F

Diameter=Maximum Distance Between Elements in a Set

Diameter=D(G,J)=D(C,J)=D(G,I)=…=5

A B

DG

E

C

F

Average Path Length

A B C D E F G

ABCDEFG

1 2 1 1 1 2 3 2 2 2 3 1 1 3 2 1 2 1 2 2 3

D(1)=8D(2)=9D(3)=4

L=(8+2x9+3x4)/(8+9+4) L=1.8

Degree CorrelationsAre Hubs Connected to Hubs?

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

Wait:are we comparing to the right thing

Compared to what?Randomized Network

AB

HI

J K

DG

EC

F AB

HI

J K

DG

EC

F AB

HI

J K

DG

EC

F

Internet

Randomized Network

Physica A 333, 529-540 (2004)

Connectivity Pattern/Randomized Z-score Connectivity Pattern/Randomized

Data

Models

After Controlling for Randomized Network

Data

Models

Fractal Networks

Mandelbrot BB

GeneratingKoch 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 natureToday we saw:Local MeasuresCentrality measures (degree, closeness, betweenness, eigenvector, page-rank)Clustering measures (Clustering, Topological Overlap or Mutual Clustering)MotifsGlobal Measures Degree Correlations, Correlation Profile.Hierarchical StructureFractal Structure

Connections Between Local and Global Measures