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

Erdös-Rényi model (1960) - Democratic - Random Pál Erdös Pál Erdös ( ) 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, (1999). WWW Expected P(k) ~ k -  Found Scale-free Network Exponential Network

INTERNET BACKBONE (Faloutsos, Faloutsos and Faloutsos, 1999) Nodes: computers, routers Links: physical lines Internet

Internet-Map

Origin of SF networks: Growth and preferential attachment Barabási & Albert, Science 286, 509 (1999) P(k) ~k -3 BA model (1) Networks continuously expand by the addition of new nodes WWW : 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 Metab-movie Protein Interactions Jeong, Tombor, Albert, Oltvai, & Barabási, Nature (2000); Jeong, Mason, Barabási &. Oltvai, Nature (2001); Wagner & Fell, Proc. R. Soc. B (2001)

Robustness Complex systems maintain their basic functions even under errors and failures (cell  mutations; Internet  router breakdowns) node failure fcfc 01 Fraction of removed nodes, f 1 S Robustness

Robustness of scale-free networks 1 S 0 1 f fcfc Attacks   3 : f c =1 (R. Cohen et al PRL, 2000) Failures Robust-SF Albert, Jeong, Barabási, Nature (2000)

Don’t forget the movie again! 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

0 km300 km100 km200 km 0 km 100 km 200 km Mobile 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 trajectories Radius of Gyration: Center of Mass:

Time dependence of human mobility Radius of Gyration:

β r =1.65±0.15 Scaling in human trajectories

β=1.75±0.15 β r =1.65±0.15 Scaling in human trajectories 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

Pu Wang Cesar Hidalgo Collaborators Marta Gonzalez