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From Complex Networks to Human Travel Patterns Albert-László Barabási
Center for Complex Networks Research Northeastern University Department of Medicine and CCSB Harvard Medical School
www.BarabasiLab.com
Erdös-Rényi model
(1960)
Connect with probability p
p=1/6 N=10 k ~ 1.5
Pál Erdös (1913-1996)
Poisson distribution
- Democratic - Random
WWW
World Wide Web
Nodes
: WWW documents
Links
: ROBOT:
collects all URL’s found in a document and follows them recursively
P(
k
) ~
k
R. Albert, H. Jeong, A-L Barabási,
Nature
,
401
130 (1999).
Internet
INTERNET BACKBONE Nodes
: computers, routers
Links
: physical lines (Faloutsos, Faloutsos and Faloutsos, 1999)
Internet-Map
BA model
Origin of SF networks: Growth and preferential attachment
(1) Networks continuously expand by the addition of new nodes WWW : addition of new documents (2) New nodes prefer to link to highly connected nodes.
WWW : linking to well known sites GROWTH: add a new node with m links PREFERENTIAL ATTACHMENT: the probability that a node connects to a node with k links is proportional to k.
(
k i
)
k i j k j
Barabási & Albert, Science 286, 509 (1999)
P(k) ~k -3
Metabolic Network Protein Interactions
Jeong, Tombor, Albert, Oltvai, & Barabási, Nature (2000); Jeong, Mason, Barabási &. Oltvai, Nature (2001); Wagner & Fell, Proc. R. Soc. B (2001)
Robustness
Robustness
Complex systems maintain their basic functions even under errors and failures (cell mutations; Internet router breakdowns) 1
S
f c
0 1
Fraction of removed nodes,
f
node failure
Robustness of scale-free networks Attacks
1
S Failures
3 :
f c
=1 (R. Cohen et al PRL, 2000) 0
f c f
Albert, Jeong, Barabási, Nature 406 1 378 (2000)
Don’t forget the movie again!
Human Motion
Brockmann, Hufnagel, Geisel Nature (2006)
Dollar Bill Motion
Brockmann, Hufnagel, Geisel Nature (2006)
A real human trajectory
Mobile Phone Users
Mobile Phone Users 0 km 100 km 200 km 300 km
Δr: jump between consecutive recorded locations.
β=1.75
± 0.15
Two possible explanations 1. Each users follows a Lévy flight 2. The difference between individuals follows a power law
Understanding individual trajectories
Center of Mass: Radius of Gyration:
Time dependence of human mobility
Radius of Gyration:
Scaling in human trajectories
β r =1.65
± 0.15
Scaling in human trajectories
β r =1.65
± 0.15
β=1.75
± 0.15
α=1.2
Relationship between exponents
Jump size distribution P(Δr)~(Δr) -β represents a convolution between *population heterogeneity P(r g )~r g -βr *Levy flight with exponent α truncated by r g
The shape of human trajectories
Collaborators
Pu Wang Marta Gonzalez Cesar Hidalgo