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Petter HolmeUmeå University, Sungkyunkwan University,Stockholm University, Institute for Future StudiesSang Hoon LeeUmeå University, Oxford University

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How can we measurenavigability?What does optimallynavigable networks looklike?

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Full informationShortest paths

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3 t 4 62 51 8 7 s 9

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3 t 4 62 51 8 7 s 9

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3 t 4 62 51 8 7 s 9

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3 t 4 62 51 8 7 s 9

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3 t 4 62 51 8 7 s 9

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Partial informationGreedy navigators

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3 t 4 62 51 8 7 s 9

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3 t 4 62 51 8 7 s 9

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3 t 4 62 51 8 7 s 9

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3 t 4 62 51 8 7 s 9

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3 t 4 62 51 8 7 s 9

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3 t 4 62 51 8 7 s 9

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3 t 4 62 51 8 7 s 9

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(Greedy navigator) navigability Avg. distanceRg = Avg. distance for greedy navigators

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(Greedy navigator) navigability Avg. distanceRg = Avg. distance for random navigators random navigators perform a random DFS

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Rg = 33% Rr = 24%

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Network N M dg d dr Rg RrBoston* 88 155 6.8 5.7 30.8 84% 19%null 8.6 3.7 23.2 43% 16%modelNew 125 217 8.3 6.8 44.4 82% 15%York*null 11.7 4.0 33.5 34% 12%modelLCM 184 194 62.8 20.6 86.2 33% 24%* from Youn, Gastner, Jeong, PRL (2008)

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Navigator essentiality

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0 –2 –41 –6 –8 ln |e| –5 2 4 –6 3 –7 –8

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1 02 –5 3 –10 4 ln |e| –5 –6 –7

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Optimizing spatial networkfor greedy navigators Fixed vertices, growing

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Boston roads

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MST

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graph distance

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Euclidean distance

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Optimizing spatial networkfor greedy navigators Fixed vertices

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Optimizing spatial networkfor greedy navigators Not fixed vertices

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0.2 BAdeviation from shortest path 0 optimized HK WS –0.2 Karate Club 2D square –0.4 1D ring –0.6 –0.8 –1 –1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 relative position f in greedy paths

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0.2deviation from shortest path BA 0 KK HK WS –0.2 Karate Club 2D square –0.4 1D ring –0.6 –0.8 –1 –1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 relative position f in GSN paths

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Thank you!SH Lee & P Holme, 2012. Exploring maps with greedynavigators. Phys. Rev. Lett. 108:128701.SH Lee & P Holme, 2012. A greedy-navigatorapproach to navigable city plans. To appear in Eur.J. Phys. Spec. Top.SH Lee & P Holme, 2012. Geometric properties ofgraph layouts optimized for greedy navigation.Under review Phys. Rev. E.