Temporal network navigation

Sang Hoon Lee (이상훈) School of Physics (물리학부), Korea Institute for Advanced Study (고등과학원)

http://newton.kias.re.kr/~lshlj82

Navigation on temporal networks

in collaboration with Petter Holme @ Sungkyunkwan University (성균관대학교)

2015년 가을 학술논문발표회 및 임시총회 [H12-St.01], 2015년 10월 23일 @

http://newton.kias.re.kr/~lshlj82

Navigation on spatial networks(Even?) simple organisms use chemotaxis to find a target.

Transport/navigation on road networks


“Which information do humans use?”

cognitive map (map in mind/brain)


Simplify! (distance/directional information)

“Which information do humans use?”

cognitive map (map in mind/brain)

l

amount of “useful” information

search timereal optimal path with

global information

random walk without any information

Information vs navigation efficiency (or “navigability”)

l



global information



cost to get info

l



global information



cost to get info

incomplete information

source

Greedy Spatial Navigation (GSN) protocol

SHL and P. Holme, Phys. Rev. Lett. 108, 128701 (2012); Phys. Rev. E 86, 067103 (2012); Eur. Phys. J.-Spec. Top. 215, 135 (2013).

target

source



target

θ1

θ2

θ1 > θ2

source



target

GSN search

source

real shortest path



target

GSN search

source

real shortest path

random search



target

GSN search

real shortest path length = d

random search path length = dr

greedy spatial navigation (GSN) path length = dg

For a given network embedded in a metric space,

real shortest path length = d

random search path length = dr

greedy spatial navigation (GSN) path length = dg

For a given network embedded in a metric space,

GSN navigability ⌫ = d/dg

random navigability ⇣ = d/dr

GSN navigabilityrandom navigability

N = |V|: number of nodes

100 road network data: https://sites.google.com/site/lshlj82/road_data_2km.zip

diverse values for GSN navigability vs clear scaling for random navigability: GSN navigability reflects the real characteristics of city structures

Navigability profile for large urban areas (2 km ✕ 2 km samples)

observe that its absence helps the large volume of trafficfrom the upper left part to avoid entering the central part toreach the lower right part, and induce to take more efficientperipheral roads. Of course, the external geographic factorssuch as rivers, tunnels, bridges, and roads with variousspeed limits are also important in practice. We take thesimplest approach and assume the geographical contextprimarily gives a sense of direction for the navigation,and neglect other effects. For future work it would beinteresting to extend our work with other informationinto other navigability functions, e.g., Bureau of PublicRoads (BPR) function [16]. We also notice that road 3 inFig. 3(e) with the largest e value (and the second largest bvalue) corresponds to the Harvard bridge across theCharles River, illustrating the case of deducing the crucialinfrastructure based solely on the geometric positions,without explicit awareness of the river.

The multiple linear regression results shown in Table IIdemonstrate that predicting e values is not plausible fromthe linear combination of those network and geometricmeasures, with low R2 values. From the same regressionanalysis on much larger Switzerland and European rail-ways, we observe even smaller R2 values estimated by the104 sampled source-target pairs for each removal of edge.Therefore, e or the Braessiness is a uniquely measured onlyby considering this greedy behavior of navigators. Finally,we investigate whether there is any correlation betweennavigability and various socioeconomic indices. We se-lected the 20 largest cities in the United States (U.S.),Europe, Asia, Latin America, and Africa, respectively,(100 cities in total), and used the MERKAARTOR program[20] and extracted a representative sample of each city (asquare of 2 km sides). First, we compared ! and " to thenumbers of vertices N, as shown in Fig. 4. There is astriking difference between those two cases, where thereis a clear scaling relationship between " and N [Fig. 4(b)],meaning that the random navigation is statistically

determined by the system sizes. In contrast, the widelyscattered points in Fig. 4(a) strongly suggests that thenumbers of vertices cannot predict ! at all, in addition tothe fact that purely topological measures cannot predict ein Table II. In this respect, the ! obviously reflects uniqueproperties of different cities with vastly different develop-mental histories. We could not find such measures (orlinear combinations of them)—e.g., population density,median resident income, fraction of public transit commut-ers, etc.—showing statistically significant correlationswith the navigability. Again this leads to the conclusionthat different cities have unique properties of navigabilityindependent of other socioeconomic factors. One exampleis the correlation between the navigability and the popula-tion change ratio of the 20 cities in the U.S. defined asthe ratio of the population change between 1960 and2010 to the population in 1960 [21]. We observe a veryweak negative correlation between ! and the ratio[R2 ¼ 0:09ð3Þ$—too weak perhaps for claiming a mean-ingful conclusion dependence.In summary, we have introduced a new routing strategy

incorporating greedy movement and memory of naviga-tors. This strategy, we believe, is a minimal model consid-ering the basic concept of human psychology fornavigation, namely, incomplete navigational informationand the memory not to be lost. From the results from real-world road and railway structures, we demonstrate theimportant difference in terms of centralities for navigationand the fact that there exists the celebrated Braess’s para-dox caused by the navigators’ behavior just equipped withthis simple strategy. From the observation of correlationprofiles for centralities in road structures, we have shownthat the importance of each element heavily depends on thedetailed layout of structures. We have focused on the finalefficiency of the routing processes in this work, but thedetailed process of GSN, e.g., the relative distance towardthe target during the routing process or the prevalence ofbacktracking related to the structural properties of roads,can be worthwhile future work. This type of tool—linkingspatial cognition, the environment, and emergent naviga-tional properties—can be helpful for urban planners andarchitects [22].This research is supported by the Swedish Research

Council and theWCU program through NRF Korea fundedby MEST R31-2008-10029 (P. H.). The authors thank

TABLE II. Coefficients for the multiple linear regression e ¼m1bþm2ðlengthÞ þm3cþm4ðkikjÞ for road networks, withsome measures defined on edges: b, the edge length, the distancec from the midpoint of edges to the centroid of vertices, and theproduct kikj of degrees of vertices attached to edges. Thestatistical significance codes are <0:05, <0:01, and <0:001.

Road Boston NYC

m1 6.902c 9.389c

m2 &4:687' 10&5a &6:141' 10&5b

m3 &1:504' 10&6 2:142' 10&5a

m4 &8:817' 10&3b &5:653' 10&3a

Multiple R2 0:2508 0:1917p value 7:784' 10&9 3:395' 10&9

a<0:05.b<0:01.c<0:001.

0

0.1

0.2

102 1030

0.4

0.2

0.6

0.8

102 103

(a) (b)

USEurope

Asia

AfricaLatin America

N N

FIG. 4 (color online). Scatter plots for the ! (a) and " (b) vs thenumber of vertices N, for the 100 large cities in the world.

PRL 108, 128701 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

23 MARCH 2012

128701-4

GSN navigability ⌫ = d/dg

random navigability ⇣ = d/dr

GSN navigability

random navigabilitySHL and P. Holme, Phys. Rev. Lett. 108, 128701 (2012).

https://sites.google.com/site/lshlj82/road_data_2km.zip

1234

time t

1

2

3

4

node-centric timeline static representation

reviews on temporal networks: P. Holme and J. Saramäki, Phys. Rep. 519, 97 (2012); P. Holme, Eur. Phys. J. B 88, 234 (2015).

Navigation on temporal networks background cartoons by Mi Jin Lee

1234

time t

1

2

3

4!




1234

time t

• only the time-respecting paths are allowed.

1

2

3

4




1234

time t


1

2

3

4

node-centric timeline static representationsource

target

source

target



1234

time t

τ1→4


1

2

3

4


target

source

target



1234

time t

τ1→4


1

2

3

4


target

source

target



1234

time t

τ1→4

τ2→4


1

2

3

4


target

source

target



1234

time t

t = t0

source: node 3

target: node 4

τ1→4

τ2→4


1

2

3

4


source

target



1234

time t

t = t0

source: node 3

target: node 4

τ1→4

τ2→4

3 steps, τ2→4

1 step, τ1→4

t = t13t = t23


1

2

3

4


source

target



1234

time t

t = t0

source: node 3

target: node 4

τ1→4

τ2→4

3 steps, τ2→4

1 step, τ1→4

t = t13t = t23


1

2

3

4


source

target



1234

time t

t = t0

source: node 3

target: node 4

τ1→4

τ2→4

3 steps, τ2→4

1 step, τ1→4

t = t13t = t23


two “null model” strategies without using any information• Saramäki-Holme (SH)’s greedy walk: following every step

J. Saramäki and P. Holme, e-print arXiv:1508.00693.

1

2

3

4


source

target



1234

time t

t = t0

source: node 3

target: node 4

τ1→4

τ2→4

3 steps, τ2→4

1 step, τ1→4

t = t13t = t23


two “null model” strategies without using any information• Saramäki-Holme (SH)’s greedy walk: following every step

J. Saramäki and P. Holme, e-print arXiv:1508.00693.• “waiting for target” (WFT): waiting for the target indefinitely

1

2

3

4


source

target



Temporal network datahttp://www.sociopatterns.org/datasets

http://www.sociopatterns.org/datasets





0

10

20

30

40

50

60

70

0 50000 100000 150000 200000 250000 300000 350000 0

20000

40000

60000

80000

100000

120000

140000

160000

num

ber o

f rea

chab

le n

odes

aver

age

time

time (sec)

reachable from node 0: our modelSH

WFTtime from node 0 this model: our model

SHWFT

“hospital” data: node 0

0

10

20

30

40

50

60

70

0 50000 100000 150000 200000 250000 300000 350000 0

20000

40000

60000

80000

100000

120000

140000

160000

num

ber o

f rea

chab

le n

odes

aver

age

time

time (sec)



SHWFT


optimal time point when the reachability is maximum for our model?

0

10

20

30

40

50

60

70

0 50000 100000 150000 200000 250000 300000 350000 0

20000

40000

60000

80000

100000

120000

140000

160000

num

ber o

f rea

chab

le n

odes

aver

age

time

time (sec)



SHWFT


number of reachable nodes(reachability): WFT > our model > SH

1/(average time) (among reachable nodes):SH > our model > WFT

optimal time point when the reachability is maximum for our model?

correlation between aggregated network measures vs navigability measures

• accumulated degree and strength (the number of contacts)

• duration of presence• time to the first contact• Goh-Barabási burstiness of nodes

(coefficient of variation): K.-I. Goh and A.-L. Barabási, EPL 81, 48002 (2008)

B ⌘ �⌧ �m⌧

�⌧ +m⌧

where �⌧ and m⌧ are the standard deviation and

mean of the interevent time ⌧ , respectively

correlation between aggregated network measures vs navigability measures

• accumulated degree and strength (the number of contacts)

• duration of presence• time to the first contact• Goh-Barabási burstiness of nodes

(coefficient of variation): K.-I. Goh and A.-L. Barabási, EPL 81, 48002 (2008)

B ⌘ �⌧ �m⌧

�⌧ +m⌧

where �⌧ and m⌧ are the standard deviation and

mean of the interevent time ⌧ , respectively

Outlook and future works

• temporal network navigation based on history to investigate the temporal correlation structure of interactions, just like spatial greedy navigation to investigate the intrinsic navigability of spatial structures

• trying to find explanatory variables related to temporal navigability!

• more (temporal network) data?

the slides in .pdf: http://www.slideshare.net/lshlj82/temporal-network-navigation

Thank you for your attention! (… and if you didn’t pay your attention, that’s entirely my fault)

http://www.slideshare.net/lshlj82/temporal-network-navigation

Science

Temporal network navigation