Exploring spatial networks with greedy navigators

Petter Holme
Petter HolmeSungkyunkwan University
Petter Holme
Umeå University, Sungkyunkwan University,
Stockholm University, Institute for Future Studies

Sang Hoon Lee
Umeå University, Oxford University
Exploring spatial networks with greedy navigators
Exploring spatial networks with greedy navigators
How can we measure
navigability?


What does optimally
navigable networks look
like?
Full information
Shortest paths
3       t               4

        6
2                       5



1       8           7
    s           9
3       t               4

        6
2                       5



1       8           7
    s           9
3       t               4

        6
2                       5



1       8           7
    s           9
3       t               4

        6
2                       5



1       8           7
    s           9
3       t               4

        6
2                       5



1       8           7
    s           9
Partial information
Greedy navigators
3       t               4

        6
2                       5



1       8           7
    s           9
3       t               4

        6
2                       5



1       8           7
    s           9
3       t               4

        6
2                       5



1       8           7
    s           9
3       t               4

        6
2                       5



1       8           7
    s           9
3       t               4

        6
2                       5



1       8           7
    s           9
3       t               4

        6
2                       5



1       8           7
    s           9
3       t               4

        6
2                       5



1       8           7
    s           9
(Greedy navigator) navigability

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

          Avg. distance
Rg =
     Avg. distance for random
     navigators

             random navigators
             perform a random DFS
Exploring spatial networks with greedy navigators
Exploring spatial networks with greedy navigators
Rg = 33%   Rr = 24%
Network N M        dg     d     dr    Rg   Rr

Boston* 88 155     6.8    5.7   30.8 84% 19%
null
                   8.6    3.7   23.2 43% 16%
model
New
         125 217   8.3    6.8   44.4 82% 15%
York*
null
                   11.7   4.0   33.5 34% 12%
model
LCM      184 194   62.8 20.6 86.2 33% 24%

* from Youn, Gastner, Jeong, PRL (2008)
Navigator essentiality
0
                  –2
                  –4
1                 –6
                  –8
                ln |e|
                   –5
    2       4      –6
        3           –7
                   –8
1           0
2
                  –5

            3     –10
    4
                ln |e|
                   –5
                  –6
                   –7
Optimizing spatial network
for greedy navigators

    Fixed vertices, growing
Boston roads
MST
graph distance
Euclidean distance
Optimizing spatial network
for greedy navigators

             Fixed vertices
Exploring spatial networks with greedy navigators
Exploring spatial networks with greedy navigators
Exploring spatial networks with greedy navigators
Optimizing spatial network
for greedy navigators

          Not fixed vertices
Exploring spatial networks with greedy navigators
Exploring spatial networks with greedy navigators
0.2
                                                              BA
deviation 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
0.2
deviation 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
Thank you!
SH Lee & P Holme, 2012. Exploring maps with greedy
navigators. Phys. Rev. Lett. 108:128701.

SH Lee & P Holme, 2012. A greedy-navigator
approach to navigable city plans. To appear in Eur.
J. Phys. Spec. Top.

SH Lee & P Holme, 2012. Geometric properties of
graph layouts optimized for greedy navigation.
Under review Phys. Rev. E.
1 of 42

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Exploring spatial networks with greedy navigators

  • 1. Petter Holme Umeå University, Sungkyunkwan University, Stockholm University, Institute for Future Studies Sang Hoon Lee Umeå University, Oxford University
  • 4. How can we measure navigability? What does optimally navigable networks look like?
  • 6. 3 t 4 6 2 5 1 8 7 s 9
  • 7. 3 t 4 6 2 5 1 8 7 s 9
  • 8. 3 t 4 6 2 5 1 8 7 s 9
  • 9. 3 t 4 6 2 5 1 8 7 s 9
  • 10. 3 t 4 6 2 5 1 8 7 s 9
  • 12. 3 t 4 6 2 5 1 8 7 s 9
  • 13. 3 t 4 6 2 5 1 8 7 s 9
  • 14. 3 t 4 6 2 5 1 8 7 s 9
  • 15. 3 t 4 6 2 5 1 8 7 s 9
  • 16. 3 t 4 6 2 5 1 8 7 s 9
  • 17. 3 t 4 6 2 5 1 8 7 s 9
  • 18. 3 t 4 6 2 5 1 8 7 s 9
  • 19. (Greedy navigator) navigability Avg. distance Rg = Avg. distance for greedy navigators
  • 20. (Greedy navigator) navigability Avg. distance Rg = Avg. distance for random navigators random navigators perform a random DFS
  • 23. Rg = 33% Rr = 24%
  • 24. Network N M dg d dr Rg Rr Boston* 88 155 6.8 5.7 30.8 84% 19% null 8.6 3.7 23.2 43% 16% model New 125 217 8.3 6.8 44.4 82% 15% York* null 11.7 4.0 33.5 34% 12% model LCM 184 194 62.8 20.6 86.2 33% 24% * from Youn, Gastner, Jeong, PRL (2008)
  • 26. 0 –2 –4 1 –6 –8 ln |e| –5 2 4 –6 3 –7 –8
  • 27. 1 0 2 –5 3 –10 4 ln |e| –5 –6 –7
  • 28. Optimizing spatial network for greedy navigators Fixed vertices, growing
  • 30. MST
  • 33. Optimizing spatial network for greedy navigators Fixed vertices
  • 37. Optimizing spatial network for greedy navigators Not fixed vertices
  • 40. 0.2 BA deviation 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
  • 41. 0.2 deviation 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
  • 42. Thank you! SH Lee & P Holme, 2012. Exploring maps with greedy navigators. Phys. Rev. Lett. 108:128701. SH Lee & P Holme, 2012. A greedy-navigator approach to navigable city plans. To appear in Eur. J. Phys. Spec. Top. SH Lee & P Holme, 2012. Geometric properties of graph layouts optimized for greedy navigation. Under review Phys. Rev. E.