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Presentation transcript:

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

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

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

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

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

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…

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

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

Airplane route network

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

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

Graph representation different granularities Internet

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

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

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….

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

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

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

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

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

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

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