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From Complex Networks
to
Human Travel Patterns
Albert-László BarabásiAlbert-László BarabásiCenter for Complex Networks ResearchCenter for Complex Networks Research
Northeastern UniversityNortheastern University
Department of Medicine and CCSBDepartment of Medicine and CCSB
Harvard Medical School Harvard Medical School
www.BarabasiLab.com
Erdös-Rényi model (1960)
- Democratic
- Random
Pál ErdösPál Erdös (1913-1996)
Connect with probability p
p=1/6 N=10
k ~ 1.5 Poisson distribution
World Wide Web
Over 10 billion documents
ROBOT: collects all URL’s found in a document and follows them recursively
Nodes: WWW documents Links: URL links
R. Albert, H. Jeong, A-L Barabási, Nature, 401 130 (1999).
Exp
ected
P(k) ~ k-
Fou
nd
Sca
le-f
ree
Netw
ork
Exp
on
en
tial
Netw
ork
INTERNET BACKBONE
(Faloutsos, Faloutsos and Faloutsos, 1999)
Nodes: computers, routers Links: physical lines
Origin of SF networks: Growth and preferential attachment
Barabási & Albert, Science 286, 509 (1999)
jj
ii k
kk
)(
P(k) ~k-3
(1) Networks continuously expand by the addition of new nodesWWW : addition of new documents
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.
(2) New nodes prefer to link to highly connected nodes.WWW : linking to well known sites
Metabolic Network Protein Interactions
Jeong, Tombor, Albert, Oltvai, & Barabási, Nature (2000); Jeong, Mason, Barabási &. Oltvai, Nature (2001); Wagner & Fell, Proc. R. Soc. B (2001)
RobustnessComplex systems maintain their basic functions even under errors
and failures (cell mutations; Internet router breakdowns)
node failure
fc
0 1Fraction of removed nodes, f
1
S
Robustness of scale-free networks
1
S
0 1f
fc
Attacks
3 : fc=1
(R. Cohen et al PRL, 2000)
Failures
Albert, Jeong, Barabási, Nature 406 378 (2000)
Two possible explanations
1. Each users follows a Lévy flight
2. The difference between individuals follows a power law
β=1.75±0.15
Δr:jump between consecutive recorded locations.
Understanding individual trajectoriesUnderstanding individual trajectories
Radius of Radius of Gyration:Gyration:
Center of Mass:Center of Mass:
Time dependence of human mobilityTime dependence of human mobility
Radius of Radius of Gyration:Gyration:
Relationship between exponentsRelationship between exponents
Jump size distribution P(Δr)~(Δr)-β represents a convolution between
*population heterogeneity P(rg)~rg-βr
*Levy flight with exponent α truncated by rg