1. Search in a Small World JIN Xiaolong Based on [1]

2. Search in a Small World2 Two important concepts Characteristic path length L where d i,j is the shortest length between nodes i and j; A global property; Clustering coefficient C where d(v i ) is the degree of node v i, e(v i ) is the number of edges existed between v i and its neighbors; A local property;

3. Search in a Small World3 Small world A graph with n nodes and m edges has a small world topology, iff: where, L random and C random are the characteristic path length and clustering coefficient averaged on all random graphs with n nodes and m edges.

4. Search in a Small World4 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 > 1;

5. Search in a Small World5 Model a small world

6. Search in a Small World6 Model a small world (2)

7. Search in a Small World7 Small world in search problems Graph coloring Time tabling Quasigroup problems

8. Search in a Small World8 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;

9. Search in a Small World9 Small world & search cost

10. Search in a Small World10 Small world & heavy-tailed distribution

11. Search in a Small World11 Reference 1. T. Walsh, Search in a Small World, in Proceedings of the Sixteenth International Joint Conference on Artificial Intelligence (IJCAI’99), 1999.