1.
Petter Holme
Umeå University, Sungkyunkwan University,
Stockholm University, Institute for Future Studies
Sang Hoon Lee
Umeå University, Oxford University

2.
How can we measure
navigability?
What does optimally
navigable networks look
like?

3.
Full information
Shortest paths

4.
3 t 4
6
2 5
1 8 7
s 9

5.
3 t 4
6
2 5
1 8 7
s 9

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.
Partial information
Greedy navigators

10.
3 t 4
6
2 5
1 8 7
s 9

11.
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.
(Greedy navigator) navigability
Avg. distance
Rg =
Avg. distance for greedy
navigators

18.
(Greedy navigator) navigability
Avg. distance
Rg =
Avg. distance for random
navigators
random navigators
perform a random DFS

19.
Rg = 33% Rr = 24%

20.
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)

21.
Navigator essentiality

22.
0
–2
–4
1 –6
–8
ln |e|
–5
2 4 –6
3 –7
–8

23.
1 0
2
–5
3 –10
4
ln |e|
–5
–6
–7

24.
Optimizing spatial network
for greedy navigators
Fixed vertices, growing

25.
Boston roads

26.
MST

27.
graph distance

28.
Euclidean distance

29.
Optimizing spatial network
for greedy navigators
Fixed vertices

30.
Optimizing spatial network
for greedy navigators
Not fixed vertices

31.
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

32.
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

33.
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.