Small-world networks

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Small-world networks
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What is it?

• Everyone talks about the small world phenomenon,
  but truly what is it? There are three landmark
  papers
• Stanley Milgram (1967)
• Duncan Watts Steve Strogatz (1998)
• Jon Kleinberg (2001 )

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Milgrams experiment

• A person P in Nebraska was given a letter to
  deliver to another person Q in Massachusetts. P
  was told about Qs address and occupation, and
  instructed to send the letter to someone she knew
  on a first-name basis in order to transmit the
  letter to the destination as fast as possible.

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Milgrams experiment

• Over many trials, the average number of
  intermediate steps in a successful chain was
  found to lie between 5 and 6.

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Milgrams experiment

• Initial success rate was very low (5). The
  follow-up experiments used some modifications of
  the original experiment.
• The outcome of the experiment led to the term
• six degrees of separation

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Other example of small world

• Small world graphs are highly clustered like
  regular lattices, yet paths of short length exist
  between random peers. Example of such graphs are
• Power grid of western US
• Collaboration graph of movie actors
• Neural network of worm C-elegans

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Watts and Strogatz 1998

• Research originally inspired by Watts' efforts
  to understand the synchronization of cricket
  chirps, which show a high degree of coordination
  over long ranges, as though the insects are being
  guided by an invisible conductor.

Disease spreads faster over a small-world
network.
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Questions not answered

• Why six degrees of separation? Any scientific
  reason? What properties do these social graphs
  have? Are there other situations in which this
  model is applicable?
• Time to reverse engineer this.

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A characterization of graphs

• Completely regular (rewiring with probability
  p0)
• Small-world graphs (p small, close to 0)
• Completely random (p1)

What is rewiring?
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Completely regular
N20 K 4 (each node has k neighbors)
High clustering coefficient and high diameter. C
3/4, L N/k
A ring lattice
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Completely random
LOW clustering coefficient and LOW diameter. C
k/n, L O(log N)
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Small world graphs
With probability p rewire each link in a regular
graph to a randomly selected node
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Clustering Coefficient and Characteristic Path
Length
p in log-scale
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Milgrams experiment revisited

• Milgrams experiment showed that
• There exist short paths in large networks that
  connect individuals
• People are able to find these short paths using a
  simple, possibly greedy, decentralized algorithm
• Small world models only take care of (a)
• What about (b)? Watts-Strogatz model is of
• little help.

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Kleinbergs question

• Watts and Strogatzs research only showed the
  existence of short paths between arbitrary pair
  of nodes. What is the guarantee that one can find
  such a path for communication?
• (If there is no algorithm for finding the short
  path, then it is not of much value!)

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Results

• Theorem 1.
• When r 0, no decentralized algorithm can find
  the short chains (even if they exist).

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Results

• Theorem 2.
• When r0, no decentralized algorithm can find
  the short chains (even if they exist). The
  expected no of hops to connect is O(n2/3).
• Uniform distribution prevents a decentralized
  algorithm from using any clue from the geometry
  of the grid.

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Kleinbergs model

• Consider a directed 2-dimensional lattice where
  each
• node u has four neighbors at lattice distance 1
• For each vertex u add q shortcuts
• choose vertex v as the destination of the
  shortcut with probability proportional to
  d(u,v)-r
• when r 0, we have uniform probabilities