Search in a Small World

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Search in a Small World

• JIN Xiaolong
• Based on 1

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Two important concepts

• Characteristic path length L
• where di,j is the
  shortest length between nodes i and j
• A global property
• Clustering coefficient C
• where
  d(vi) is the degree of node vi, e(vi) is the
  number of edges existed between vi and its
  neighbors
• A local property

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Small world

• A graph with n nodes and m edges has a small
  world topology, iff
• where, Lrandom and Crandom are the
  characteristic path length and clustering
  coefficient averaged on all random graphs with n
  nodes and m edges.

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Small world (2)

• Generally, a small world topology appears in a
  sparse and connected graph
• A quantitative measurement, proximity ratio
• A small world topology requires µ gt 1

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Model a small world
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Model a small world (2)
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Small world in search problems

• Graph coloring
• Time tabling
• Quasigroup problems

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Small world search cost (2)

• In a graph with a small world topology, local
  properties can be bad predicators for global
  property
• Heuristics often use local properties to guide
  the search for a (global) solution
• Because of this mismatch, a small world topology
  may mislead heuristics and make search problems
  hard to solve

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Small world search cost
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Small world heavy-tailed distribution
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Reference

• 1. T. Walsh, Search in a Small World, in
  Proceedings of the Sixteenth International Joint
  Conference on Artificial Intelligence (IJCAI99),
  1999.