Transcript Slide 1

The structure of the Internet
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?
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
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
100
90
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
100
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
SCIENCE CITATION INDEX
Nodes: papers
Links: citations
25
Witten-Sander
PRL 1981
1736 PRL papers (1988)
2212
P(k) ~k-
( = 3)
(S. Redner, 1998)
Sex-web
Nodes: people (Females; Males)
Links: sexual relationships
4781 Swedes; 18-74;
59% response rate.
Liljeros et al. Nature 2001
Web power-laws
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
(1) GROWTH :
Scale-free model
At 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 ki of that node
ki
 ( ki ) 
 jk j
P(k) ~k-3
A.-L.Barabási, R. Albert, Science 286, 509 (1999)
The Faloutsos Graph
node degree for AS20000102.m
4
10
3
10
2
10
1
10
0
10
0
10
1
10
2
10
3
10
4
10
The Internet Topology as a
Jellyfish
Shells: 1
3
2
Core
 Core: High-degree clique
 Shell: adjacent nodes of
previous shell, except 1degree nodes
 1-degree nodes: shown
hanging
 The denser the 1-degree
node population the
longer the stem