Optimization Based Modeling of Social Network

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Optimization Based Modeling of Social Network

• Yong-Yeol Ahn, Hawoong Jeong

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Outline

• About real networks and models
• Motivation
• Simulation method
• Result
• Conclusion

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Real Complex Networks

• Social networks
• Acquaintance, scientific collaboration, actor,
  bbs, etc.
• Internet, WWW, e-mail, other communication
  networks

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Real Complex Networks

• Biological networks
• Metabolic network
• Genetic network
• Protein interaction network
• Neuronal network

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Basic Concepts of Network
Degree 3
Links
A shortest path with path length3
(Equivalent with 3 clicks in WWW)
Nodes
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Clustering Coefficient

• Clustering coefficient for a node represent how
  many links are there between neighbors
• Clustering coefficient for a network is the
  average of all nodess clustering coefficient

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Clustering Coefficient
A clique or a community C1
C0
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Clustering Coefficient
Triangle ? the building block.
Alternative definition of clustering coefficient
3 x of triangle
C
of connected triples
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Real Networks Universal Characters

• Short path length
• High clustering
• Large inhomogeneity (power-law degree
  distribution)

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Modeling Real Networks

• Static network model
• Erdös-Rényi model(random network)

Connect All pairs of nodes with probability p
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Erdös-Rényi Model

• Randomness ? short path length
• Homogeneous model

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Modeling Real Networks

• Static network model
• Watts-Strogatz model (small world)

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Modeling Real Networks

• Watts-Strogatz model

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Small World Network Model

• Randomness ? short path length
• Regularity ? high clustering
• Balance between regularity and randomness

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Modeling Real Networks

• Growing network model
• BA model
• From the power law degree distribution of real
  networks
• Many models after BA model adopted the growing
  scheme

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Network Models BA Model

• Growing
• New nodes and links are added continuously
• Linear preferential attachment
• New nodes make links with preferential attachment
  rule
• Rule Riches get richer

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Scale-free Network Model

• Scale-free network model
• Hub and power-law degree distribution ?
  inhomogeneity
• Network is growing
• and inhomogeneous

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New Scheme Optimization
BA model says A network is growing
New models say The evolution is more important
than growth. Lets ignore the growth (Mathias
et al.)
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Growth and Evolution
Growth Addition of nodes Evolution Rewiring
of links

• WWW is growing exponentially
• Rewiring in WWW is faster than growth
• Bacteria ? Human (Growth of biological networks)
• Origin of species (Numerous rewiring in
  biological networks)

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Evolutionary Pressure

• So, the rewiring occur randomly?
• ? No.
• Biological networks
• Natural selection
• Artificial networks(electrical circuit,)
• Cost, High performances

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New Design Optimization Models

• Origin of biological networks and man-made
  networks
• Timescale of link dynamics vs. Timescale of node
  dynamics
• ? Take a snapshot
• Growth ? rewiring, evolution

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Examples of Optimization

• In biosystems
• Metabolic networks path length conservation
• Allometric scaling
• In artificial systems
• JAVA class network(A structure of computer
  program)
• Electric circuit

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Optimization Scheme

• How to model the natural selection and
  optimization?
• ? Nature want to enlarge networks efficiency
  while want to cut down cost
• So,
• High efficiency ? short path length
  (Information flow)
• Low cost ? fewer links
• Energy p L (1-p)E
• (pparameter, Lpath length, E expense, cost)

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Star Network

• Trivial case optimizing only average path length

To shorten path length
makes a hub
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Result of Optimization Model

• Power law degree distribution in some range of p
  (parameter)

(Cancho and Sole)
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Our Motivation

• Real networks have large clustering coefficient
  and community structures
• Then,
• What kind of network will we get, if we maximize
  a networks clustering cofficient?

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Method

• Greedy algorithm
• Choose a link and rewire it randomly
• If energy decreases, keep it
• If energy increases, discard it
• We calculate with or without connection
  constraint

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Method Supplement
This link is weak under our method
Strong link
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Energy Optimization

• Maximizing clustering coefficient
• Energy 1 - C (C Clustering coefficient)
• We try to maximize clustering coefficient
• Generalized form
• Energy p(1-C) (1-p)d
• P balances contributions from C and d
• We try to maximize clustering and to minimize
  normalized vertex-vertex distance

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ResultsClustering Only (NotConnected)
Scale free network with exponent
2.2 (N10000,L20000) Clustering coeff. 0.83
P(k)
Degree
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ResultsClustering Only (NotConnected)
Structure of the network. N300, L600,
Clustering coeff. 0.9
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Results Clustering Only(connected)
Exponent 2.9 (N10000,L20000) Clustering
coeff. 0.79
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Results Clustering Only(connected)
Structure of the network
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Results Clustering and Distance
Only by path length
p0
Only by clustering coefficient
p1
p0.1
We can observe large differences in topology
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Discussion

• Lets see social networks
• Can we define cost in social networks?
• Can we define efficiency in social networks?
• ? Social networks are different from biological
  and artificial networks.

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Discussion

• Functional networks Metabolic network,
  Electrical circuit network, ..
• ? global
• Non-functional network Social networks, e-mail
  network, ..
• ? Local

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Discussion

• Creation and deletion of a link in non-functional
  network.
• Creation of link ? through friends
• Deletion of link ? through out of sight, out of
  mind

? Simplified to rewiring
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Discussion

• Two forces
• Make triangles!
• Make hubs!

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Discussion

• The two forces make power-law degree distribution
• If we add average path length in energy function,
  large hubs result.

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Conclusion

• We categorize networks into two groups
• We explain the meaning of clustering-driving
  scheme
• With clustering optimization, we get highly
  clustered scale-free network