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