1. The structure of the Internet

2. The Internet as a graph Remember: the Internet is a collection of networks called autonomous systems (ASs) The Internet graph: –The AS graph Nodes: Ass, links: AS peering –The router level graph Nodes: routers, links: fibers, cables,MW channels, etc. How does it looks like?

3. Random graphs in Mathematics The Erdös-Rényi model Generation: –create n nodes. –each possible link is added with probability p. Number of links: np If we want to keep the number of links linear, what happen to p as n  ? Poisson distribution

4. The Waxman model Generation –Spread n nodes on a large enough grid. –Pick a link uar and add it with prob. that exponentially decrease with its length –Stop if enough links Heavily used in the 90s

6. 1999 The Faloutsos brothers Measured the Internet AS and router graphs. Mine, she looks different! Notre Dame Looked at complex system graphs: social relationship, actors, neurons, WWW Suggested a dynamic generation model

8. SCIENCE CITATION INDEX (  = 3) Nodes: papers Links: citations (S. Redner, 1998) P(k) ~k -  2212 25 1736 PRL papers (1988) Witten-Sander PRL 1981

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

10. Web power-laws

11. SCALE-FREE NETWORKS (1) The number of nodes (N) is NOT fixed. Networks continuously expand by the addition of new nodes Examples: WWW : addition of new documents Citation : publication of new papers (2) The attachment is NOT uniform. A node is linked with higher probability to a node that already has a large number of links. Examples : WWW : new documents link to well known sites (CNN, YAHOO, NewYork Times, etc) Citation : well cited papers are more likely to be cited again

12. Scale-free model (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 A.-L.Barabási, R. Albert, Science 286, 509 (1999) P(k) ~k -3

13. The Faloutsos Graph

15. The Internet Topology as a Jellyfish  Core: High-degree clique  Shell: adjacent nodes of previous shell, except 1- degree nodes  1-degree nodes: shown hanging  The denser the 1-degree node population the longer the stem Core Shells: 1 2 3