Complex Human Mobility Dynamics on a Network

Michael Szell, Giovanni Petri, Roberta Sinatra, Vito Latora, Stefan Thurner

Section for Science of Complex SystemsMedical University of Vienna

www.complex-systems.meduniwien.ac.at

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Why study human mobility?

• Spread of epidemics

• Urban planning (railway, underground, airline)

• Crowd dynamics

• Geomarketing

• Spread of computer viruses (WIFI, Bluetooth)

Barthelemy, arXiv:1010.0302 (2010)Castellano et al, Rev Mod Phys 81, 591-646 (2009)

Pastor-Satorras and Vespignani, PRL 86, 3200-3203 (2001)Wang et al, Science 324, 1071-1075 (2009)

Helbing et al, PRE 75, 046109 (2007)

Statistical physics of social dynamics

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Measuring mobility

Large-scale datasets: mobile phones, dollar bills, subway

Brockmann et al, Nature 439, 462-465 (2006)González et al, Nature 453, 779-782 (2008)

Roth et al, arXiv:1001.4915 (2010)Song et al, Science 327, 1018-1021 (2010)

Koelbl and Helbing, New J of Phys 5, 48 (2003)

• Topology often a spatial network

• Scale-free spatial step distributions

• Scale-free waiting time distributions

• High predictability + anomalous diffusion

Universal laws? Diffusion model?

3

Establishing a socio-economic laboratory

Bainbridge, Science 317, 472 (2007)Szell and Thurner, Social Networks 32, 313-329 (2010)

Szell et al, PNAS 107, 1363-13641 (2010)

www.pardus.at

400,000 participants live an alternative life, in an online society interacting with others

• trading• socializing• conflicting

All data available!

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Mobility on a network

Pajek

• 400 Nodes

• Planar

• Diameter 27

• Lattice-like

Day-to-day mobility of 2000 participants over 500 days

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Mobility on a network

Pajek

Metric defined by shortest paths

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Restrictions

• Number of actions

• Travel cost

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Diffusion and MSD

MSD =�r2(t)

�

Mean Squared Displacement

ν = limt→∞

d

dt(MSD)

ν

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Human mobility is highly subdiffusive

MSD =�r2(t)

�

Mean Squared Displacement

ν = limt→∞

d

dt(MSD)

ν = 0.4 < 1

Similar in mobile phone studySong et al, Nature Physics 6, 818-823 (2010)

100 101 102100

101

102

MSD ~ t0.4

MSD = t1

t (days)

MSD

Random WalkersHumans

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Waiting times and spatial steps

P (∆r) ∼ e−1/3∆r ?P (∆t) ∼ ∆t−1−β

0 5 10 15 20 2510 6

10 4

10 2

100

Spatial step r

P( r)

slope = 1/3

100 101 102 10310 6

10 4

10 2

100

Waiting time t

P( t)

slope = 2.16

β ≈ 1.16

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M

100+

80

60

40

20

0

Transition probabilities and passage times

Probability to jump from node i to node j

Mean time to walk from node i to node j

Distance from node i to node jD = (dij) =

Q = (qij) =

M = (mij) =

Q

1

10 2

10 4

10 6

10 8

0

D

0

5

10

15

20

25

How do they relate?

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100 101 102 103 104100

101

102

slope = 1/3

q ij

mij

Transition probabilities and passage times

Probability to jump from node i to node j

Mean time to walk from node i to node j

Distance from node i to node jD = (dij) =

Q = (qij) =

M = (mij) =

0 5 10 15 20 25100

101

102

103

Distance dij

<q ij><mij>

Why saturation?Mobility law ? m ∼ q−13

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Two scales of mobility

Pajek

1) Intra-cluster mobility: daily routine, home work

2) Inter-cluster mobility: extraordinary events, war

↔

dominates

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Comparison with mobile phone study

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

Mobile phone users Pardus users

Srand = log2 U

Sunc = −400�

i=1

pi log2 pi

Song et al, Science 327, 1018-1021 (2010)

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Model: Return to previous node

Draw spatial step from exponential, random-walk until distance reached

Return to previous node 0 5 10 15 20 2510 6

10 4

10 2

100

Spatial step r

P( r)

slope = 1/3

q

1− q

0 ≤ q < 1(where )

Reproduces properties best (so far)

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Summary

• Establish a socio-economic laboratory

• Subdiffusive mobility of humans, as in phone study

• Mobility laws

• Two scales: daily routine versus events

michael.szell@meduniwien.ac.at

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