1. Stefano Boccaletti Complex networks in science and society *Istituto Nazionale di Ottica Applicata - Largo E. Fermi, 6 - 50125 Florence, ITALY *CNR-Istituto dei Sistemi Complessi * MIND- Mediterranean Institute for Nonlinear Dynamics Coworkers: Dong-Uk Hwang, Mario Chavez, Andreas Amann,Vito latora Hector Mancini, Jean Bragard, Louis Pecora, Juergen Kurths Dedicated to the memory of Carlos Pérez Garcia PAMPLONA 2005

2. Summary WHAT IS A NETWORK? WHAT IS A COMPLEX NETWORK? THE STRUCTURE OF COMPLEX NETWORKS THE MODELS OF COMPLEX NETWORKS

3. Do you want to know more?  S.Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang COMPLEX NETWORKS: STRUCTURE AND DYNAMICS 212 pages, 856 References TO APPEAR SOON IN PHYSICS REPORTS For preprints write to stefano@ino.it

4. Society Nodes: individuals Links: social relationship (family/work/friendship/etc.) S. Milgram (1967) John Guare Six Degrees of Separation Social networks: Many individuals with diverse social interactions between them.

5. Communication networks The Earth is developing an electronic nervous system, a network with diverse nodes and links are -computers -routers -satellites -phone lines -TV cables -EM waves

6. INTERNET BACKBONE

7. Erdös-Rényi model (1960) Pál Erdös Pál Erdös (1913-1996) Connect with probability p Poisson distribution

8. ARE COMPLEX NETWORKS REALLY RANDOM?

9. Road and Airline networks Poisson distribution Exponential Network Power-law distribution Scale-free Network Airlines

10. SCIENCE CITATION INDEX Nodes: papers Links: citations P(k) ~k -  2212 25 Witten-Sander PRL 1981

11. SCIENCE COAUTHORSHIP Nodes: scientist (authors) Links: write paper together

12. ACTOR CONNECTIVITIES Nodes: actors Links: cast jointly Days of Thunder (1990) Far and Away (1992) Eyes Wide Shut (1999) N = 212,250 actors  k  = 28.78 P(k) ~k -   =2.3

13. Centrality: Why Kevin Bacon? Measure the average distance between Kevin Bacon and all other actors. No. of movies : 46 No. of actors : 1811 Average separation: 2.79 Kevin Bacon Is Kevin Bacon the most connected actor? NO! 876 Kevin Bacon 2.786981 46 1811 Bacon-list

14. Rod Steiger Martin Sheen Donald Pleasence #1 #2 #3 #876 Kevin Bacon Bacon-map

15. FOOD WEBS R.J. Williams, N.D. Martinez Nature (2000) Nodes: trophic species Links: trophic interactions

16. SEX WEBS Nodes: people (Females; Males) Links: sexual relationships 4781 Swedes; 18-74; 59% response rate. Liljeros et al. Nature 2001

17. Metabolic Networks I Nodes : chemicals (substrates) Links : bio-chemical reactions

18. Metabolic Networks II ArchaeaBacteriaEukaryotes Organisms from all three domains of life are scale-free networks! H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, and A.L. Barabasi, Nature, 407 651 (2000)

19. Protein networks I Nodes : proteins Links : physical interactions (binding) P. Uetz, et al. Nature 403, 623-7 (2000).

20. Protein networks II H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411, 41-42 (2001)

21. Nature 408 307 (2000) … “One way to understand the p53 network is to compare it to the Internet. The cell, like the Internet, appears to be a ‘scale-free network’.”

22. p53 network (mammals)

24. Watts-Strogatz Model C(p) : clustering coeff. L(p) : average path length (Watts and Strogatz, Nature 393, 440 (1998))

25. BA - Scale-free model A.-L.Barabási, R. Albert, Science 286, 509 (1999) (1) GROWTH : A t every timestep we add a new node with m edges (connected to the nodes already present in the system). (2) PREFERENTIAL ATTACHMENT : The probability Π that a new node will be connected to node i depends on the connectivity k i of that node P(k) ~k -3

26. 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

27. Achilles’ Heel of complex networks Internet failure attack Achilles Heel R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000)

28. Yeast protein network - lethality and topological position - Highly connected proteins are more essential (lethal)... Prot- robustness H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411, 41-42 (2001)