3. SMALL WORLDS

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3. SMALL WORLDS

• The Watts-Strogatz model

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Watts-Strogatz, Nature 1998

• Small world the average shortest path length in
  a real network is small
• Six degrees of separation (Milgram, 1967)
• Local neighborhood long-range friends
• A random graph is a small world

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Networks in nature (empirical observations)
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Model proposed

• Crossover from regular lattices to random graphs
• Tunable
• Small world network with (simultaneously)
• Small average shortest path
• Large clustering coefficient (not obeyed by RG)

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Two ways of constructing
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Original model

• Each node has Kgt4 nearest neighbors (local)
• Probability p of rewiring to randomly chosen
  nodes
• p small regular lattice
• p large classical random graph

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p0 Ordered lattice
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p1 Random graph
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• Small shortest path means small clustering?
• Large shortest path means large clustering?
• They discovered there exists a broad region
• Fast decrease of mean distance
• Constant clustering

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Average shortest path

• Rapid drop of l, due to the appearance of
  short-cuts between nodes
• l starts to decrease when pgt2/NK (existence of
  one short cut)

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• The value of p at which we should expect the
  transtion depends on N
• There will exist a crossover value of the system
  size

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Scaling

• Scaling hypothesis

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NN(p)
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Crossover length
d dimension of the original regular lattice
for the 1-d ring
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Crossover length on p
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General scaling form

• Depends on 3 variables, entirely determined by a
  single scalar function.
• Not an easy task

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Mean-field results

• Newman-Moore-Watts

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Smallest-world network
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• L nodes connected by L links of unit length
• Central node with short-cuts with probability p,
  of length ½
• p0 lL/4
• p1 l1

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Distribution of shortest paths

• Can be computed exactly
• In the limit L-gt?, p-gt0, but ?pL constant. zl/L
  

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different values of pL
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Average shortest path length
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Clustering coefficient

• How C depends on p?
• New definition
• C(p) 3xnumber of triangles / number of
  connected triples
• C(p) computed analytically for the original
  model

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Degree distribution

• p0 delta-function
• pgt0 broadens the distribution
• Edges left in place with probability (1-p)
• Edges rewired towards i with probability 1/N

notes
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only one edge is rewired exponential decay, all
nodes have similar number of links
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Spectrum

• ?(?) depends on K
• p0 regular lattice ? ?(?) has singularities
• p grows ? singularities broaden
• p-gt1 ? semicircle law

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3rd moment is high clustering, large number of
triangles