From Complex Networks to Human Travel Patterns Albert-László Barabási Center for Complex Networks Research Northeastern University Department of Medicine.

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)

Don’t forget the

movie again!

Don’t forget the

movie again!

Human MotionHuman Motion

Brockmann, Hufnagel, Geisel Nature (2006)

Dollar Bill MotionDollar Bill Motion

Brockmann, Hufnagel, Geisel Nature (2006)

A real human trajectory

Mobile Phone Users

0 km 300 km100 km 200 km

0 k

m10

0 k

m20

0 k

mMobile Phone Users

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:

βr=1.65±0.15

Scaling in human trajectoriesScaling in human trajectories

β=1.75±0.15βr=1.65±0.15

Scaling in human trajectoriesScaling in human trajectories

α=1.2

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

The shape of human trajectoriesThe shape of human trajectories

Pu Wang Cesar Hidalgo

CollaboratorsCollaborators

Marta Gonzalez

www.BarabasiLab.com