1. Complex Networks First Lecture TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA

2. I. A few examples of Complex Networks II. Basic concepts of graph theory and network theory III. Models IV. Communities Program

3. Two main classes Natural systems: Biological networks: genes, proteins… Foodwebs Social networks Infrastructure networks: Virtual: web, email, P2P Physical: Internet, power grids, transport…

4. protein-gene interactions protein-protein interactions PROTEOME GENOME Citrate Cycle METABOLISM Bio-chemical reactions

5. Metabolic Network Nodes: proteins Links: interactions Protein Interactions Nodes: metabolites Links:chemical reactions

6. Scientific collaboration network Nodes: scientists Links: co-authored papers Weights: depending on number of co-authored papers number of authors of each paper number of citations…

7. Actors collaboration network Nodes: actors Links: co-starred movies

8. World airport network complete IATA database V = 3100 airports E = 17182 weighted edges w ij #seats / (time scale) > 99% of total traffic

9. Airplane route network

10. Meta-population networks City a City j City i Each node: internal structure Links: transport/traffic

11. Computers (routers) Satellites Modems Phone cables Optic fibers EM waves Internet

12. Graph representation different granularities Internet

13. Virtual network to find and share informations web pages hyperlinks The World-Wide-Web CRAWLS

14. Sampling issues social networks: various samplings/networks transportation network: reliable data biological networks: incomplete samplings Internet: various (incomplete) mapping processes WWW: regular crawls … possibility of introducing biases in the measured network characteristics

15. Networks characteristics Networks: of very different origins Do they have anything in common? Possibility to find common properties? the abstract character of the graph representation and graph theory allow to answer….

16. Social networks: Milgram’s experiment Milgram, Psych Today 2, 60 (1967) Dodds et al., Science 301, 827 (2003) “Six degrees of separation” SMALL-WORLD CHARACTER

17. Small-world properties Average number of nodes within a distance l Scientific collaborations Internet

18. Clustering coefficient 1 2 3 n Higher probability to be connected Clustering: My friends will know each other with high probability (typical example: social networks) Empirically: large clustering coefficients

19. Topological heterogeneity Statistical analysis of centrality measures: P(k)=N k /N=probability that a randomly chosen node has degree k also: P(b), P(w)…. Two broad classes homogeneous networks: light tails heterogeneous networks: skewed, heavy tails

20. Topological heterogeneity Statistical analysis of centrality measures Broad degree distributions Power-law tails P(k) ~ k -   typically 2<  <3

21. Topological heterogeneity Statistical analysis of centrality measures: Poisson vs. Power-law log-scale linear scale

22. Exp. vs. Scale-Free Poisson distribution Exponential Network Power-law distribution Scale-free Network