The Natural Science of Networks

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Networks, Complexity and Economic Development

• Characterizing the Structure of Networks
• Cesar A. Hidalgo PhD

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WWW
Over 3 billion documents
Exponential Network
Scale-free Network
R. Albert, H. Jeong, A-L Barabasi, Nature, 401
130 (1999).
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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.
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Local Measures
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CENTRALITY MEASURES Measure the importance of
a node in a network.
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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
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Most Connected Actors in Hollywood (measured in
the late 90s)
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
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A-L Barabasi, Linked, 2002
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DEGREE CENTRALITY
K number of links
Where Aij 1 if nodes i and j are connected and
0 otherwise
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BETWENNESS CENTRALITY
BC number of shortest Paths that go through
a node.
N11
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CLOSENESS CENTRALITY
C Average Distance to neighbors
N11
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EIGENVECTOR CENTRALITY
Consider the Adjacency Matrix Aij 1 if node i
is connected to node jand 0 otherwise. Consider
the eigenvalue problem Axlx Then the
eigenvector centrality of a node is defined
as where l is the largest eigenvalue
associated with A.
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PAGE RANK
PRProbability that a random walker with
interspersed Jumps would visit that node. PREach
page votes forits neighbors.
K
G
B
A
H
E
F
J
C
I
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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))
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PAGE RANK
PRProbability that a random Walker would visit
that node. PREach page votes forits neighbors.
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CLUSTERING MEASURES Measure the density of a
group of nodes in a Network
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Clustering Coefficient
Ci2D/k(k-1)
CA2/121/6
CC2/21
CE4/62/3
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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))
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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
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MOTIFS
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Motifs
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Structural Equivalence
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Global Measures
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The Distribution of any of the previously
introduced measures
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Giant Component
Components
SNumberOfNodesInGiantComponent/TotalNumberOfNodes
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Diameter
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DiameterMaximum Distance Between Elements in a
Set DiameterD(G,J)D(C,J)D(G,I)5
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Average Path Length
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Degree CorrelationsAre Hubs Connected to Hubs?
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Phys. Rev. Lett. 87, 258701 (2001)
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Waitare we comparing to the right thing
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Compared to what?

• Randomized Network

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Physica A 333, 529-540 (2004)
Randomized Network
Internet
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Connectivity Pattern/Randomized
Z-score Connectivity Pattern/Randomized
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After Controlling for Randomized Network
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Fractal Networks
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Mandelbrot BB
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Generating Koch Curve
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White Noise
Pink Noise
Brown Noise
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Attack Tolerance
Non Fractal Network
Fractal Network
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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