networks, complexity and economic development characterizing the structure of networks cesar a....
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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
le-f
ree N
etw
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.
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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