Modeling the Internet Topology

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Modeling the Internet Topology
Zhengping Fan
Department of Electronic Engineering City
University of Hong Kong, China
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OUTLINE

• Introduction and Motivation
• A Real Map of the Internet
• Modeling the Internet Using the
  Multi-local-world (MLW)
• Model
• Conclusion and Perspective

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Introduction and Motivation

• The Internet is very important in our daily life!
• It can provide us three basic services

• Telnet allows communication with remote
  computers
• FTP file transfer protocol
• Email communication software

telnet FTP email
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Introduction and Motivation (Cont.)

• The Internet topology is fundamental for our
  understanding of the Internet.

• Design Efficient Protocols
• Create Efficient Searching Algorithms
• Prevent the Virus Spreading in the Internet

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Introduction and Motivation (Cont.)

• Graph representation
• Router-level modeling
• nodes are routers
• links are one-hop IP connectivity
• Domain- (AS-) level modeling
• nodes are domains
• links are peering relationships

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Introduction and Motivation (Cont.)

• Waxman model
• Locality (The farther apart the two nodes are,
  the less likely they will be connected. )

• Transit-Stub Model

• Transit-Stub Model can capture the hierarchical
  feature of the Internet.

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Introduction and Motivation (Cont.)

• Question Can the Waxman model and the
  Transit-stub model be suitable to model the
  Internet topology?

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A Real Map of the Internet
Border routers
Prefix AS path d C B A
Each entry in a BGP routing table lists the
path of ASs used to reach a destination prefix,
and thus each entry implicitly lists AS
connectivity information.
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A Real Map of the Internet (Cont.)

• Faloutsos (1999)
• degree sequence follows a power-law form
  with

Data collected by RouteViews
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A Real Map of the Internet (Cont.)
Statistical data of the Internet (1997-1999)
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A Real Map of the Internet (Cont.)

• N number of nodes
• E number of edges
• average degree
• average clustering coefficient
• average distance

c(i) the ratio between the number of connections
among the k neighbors of a given node i and its
maximum possible value.
Distance d(n,m) between two nodes n and m
number of nodes in the shortest paths connecting
them. Average distance L average over all
d(n,m)
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Modeling the Internet

• The degree distribution in the Waxman model and
  Transit-stub model follows a Poisson
  distribution.

• Neither the Waxman model nor the Transit-stub
  model can be used to model the AS-level Internet
  topology.

• To describe the scale-free statistical feature,
  Barabasi-Albert (BA) Model was proposed in 1999.

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Modeling the Internet (Cont.)
BA model includes two ingredients

• Growth Starting with a small number (m0) of
  nodes, add a new node with m links at every step
• Preferential attachment Node i whose
  connectivity is k(i) receives a new link from the
  newly added node with probability of

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Modeling the Internet (Cont.)

• Features of the BA Model
• Connectivity distribution in a power-law form,
  with r 3
• Non-homogeneous nature
• a few nodes have large numbers of
  connections but most others have much less.

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Modeling the Internet (Cont.)
The BA Model can describe scale-free networks in
power law with r 3, but the Internet network
does not satisfy r 3
The EBA model (Albert and Barabasi 2000) --
(i) Add new links between existing nodes
With probability ( ) new links
are added into the network one end of the link
is chosen at random, and the other end is
selected with probability
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Modeling the Internet (Cont.)
(ii) Re-wiring With probability links
are rewired First, a node with a link is
selected at random. Then, this link is replaced
with a new link that connects node to node
which is chosen with probability
(iii) Incremental growth With probability
, a new node is added into the network The
new node has new links to the already
existing nodes in the network with probability
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Limitation of BA and EBA Models
Modeling the Internet (Cont.)
Preferential Attachments
OR

• BA and EBA models have a global preferential
  attachment
• However, The Internet should have local
  preferential attachment probabilities since the
  localization property exists in the Internet.

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Modeling the Internet (Cont.)

• Limitations of using BGP tables to infer the
  Internet topology

• Each router can only see the Internet
  connectivity from its own limited view.

• Infrastructure addresses may not be advertised
• Does not require to be announced publicly
• Security concerns

So, some nodes and links will be lost.
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Modeling the Internet (Cont.)
Some additional databases besides RouteViews

• Route Servers (RS) Router servers are routers
  made publicly accessible by some ISP networks to
  help troubleshoot network problems.
• Routing Looking Glass (RLG) Looking Glass sites
  provide only the BGP peering relationships for an
  individual ISP, and not the AS_PATH information.
  Hence, can be used for constructing local AS
  views only.

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Modeling the Internet (Cont.)

• Internet Routing Registry (IRR) Maintains
  individual ISPs routing information in several
  public repositories to co-ordinate global routing
  policy. Two of the biggest IRR databases
  available are the ones maintained by the Routing
  Arbiter Project (RADB) and by Reseaux IP
  Europeens (RIPE). Here only RIPE is used.

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Modeling the Internet (Cont.)
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Modeling the Internet (Cont.)
yaxb where a10.9,b16700
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Modeling the Internet (Cont.)
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Modeling the Internet (Cont.)
yaxb, where a82.3,b46700
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Modeling the Internet (Cont.)
on Jan.1,2004
on Mar.30,2005
Power-law distribution
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Modeling the Internet (Cont.)
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Modeling the Internet (Cont.)
Internet is not only a scale-free network,but
also a small-world network.
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Modeling the Internet (Cont.)

• Internet consists of many sub-networks.

• The connections within the same sub-network are
  compact.

• The connections between different sub-networks
  are sparse.

• A sub-network is called local-world.

• AS birth, Link birth/death, local-world birth

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Our MLW Model

• Multi-local-world (MLW) model is proposed.

Start with isolated local-worlds, with
nodes and links in each local-world
Example Start with local-worlds (A,
B, C), with nodes (black circles) and
links in each
local-world Each local-world has a unique
identifier (red circle)
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Our MLW Model
At each step, perform one of the five operations

(i) With probability p a new local-world is
created, which contains nodes and links.
Meanwhile, a unique identifier is generated for
this new local-world.
Local world D is created with probability p
(with nodes (black circles) and
links)
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Our MLW Model
(ii) With probability q, a new node is added to
an existing local-world, which has links
with the nodes within the same local-world
First, a local-world is selected at
random. Then, a node to which the new node
connects in the local-world is chosen with
probability
(1)
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Our MLW Model
(Step (ii) continued)
In (1), is the -th local-world in which
node locates, and the parameter represents
the attractiveness of node which is used to
govern the probability for young nodes to get
new links. This process is repeated
times.
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Our MLW Model(Step (ii) continued)
Example (continued) A new node joins the
network. First, it selects the local-world C
where it will locate, and then connects an
existing node ( ) in this local-world
with preferential attachment probability given by
(1)
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Our MLW Model
(iii) With probability r, links are added to
a chosen local-world First, a local-world
is selected at random. Then, one end of a link
is chosen at random, while the other end of the
link is selected by (1) This process is repeated
times
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Our MLW Model(Step (iii) continued)
Example First, local-world C is chosen at
random. Then, links are added to
this local-world. One end of a link is selected
at random, while the other end of the link is
chosen with a probability given by (1)
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Our MLW Model
(iv) With probability s, links are deleted
within a chosen local-world First, a
local-world is selected at random. Then,
one end of a link is chosen at random, while the
other end of the link is selected with
probability


(2)
where represents the number of nodes
within the -th local-world, and
is determined by (1) This process is repeated
times.
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Our MLW Model (Step (iv) continued)
Example First, local-world C is chosen at
random. Then, link is deleted within
this chosen local-world. An end of the link is
selected at random, while the other end of the
link is chosen with probability given by (2)
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Our MLW Model
(v) With probability u, a selected local-world
has links with the other existing
local-worlds First, randomly select a
local-world and a node in this local-world with
probability given by (1). Then, the selected
node is acted as one end of a link, while the
other node of the link, which is in another
local-world chosen at random, is selected with
probability given by (1). This process is
repeated times.
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Our MLW Model (Step (v) continued)
Example Depending on the probability u,
link is added between two nodes in two
different local-worlds. Both ends of the link
are chosen with preferential attachment according
to a probability given by (1)
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Our MLW Model
In this MLW model, parameters satisfy
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Our MLW Model
Degree Distribution in the MLW model follows a
power-law form
Here
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Our MLW Model Special Cases
Case A If
then, the network has only one local-world, with
power-law exponent
So, the BA model is a special case of this MLW
model.
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Our MLW Model Special Cases
Case B If
Then, the power-law exponent is
This indicates that adding links between two
existing nodes in a network has important impact
on the scale-free characteristic of this kind of
evolving networks.
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Our MLW Model Special Cases
Case B (continued)
If, furthermore,
Then, the power-law exponent is fixed
This indicates that the attractiveness of nodes
plays a very important role in the evolution of
this kind of networks
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Our MLW Model
Comparison Results Three Different Models
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Conclusion and Perspective

• A Multi-local-world (MLW) model has been proposed
  to model the Internet topology.
• It has been shown that our MLW model is much
  better than BA model and EBA model when used to
  model the Internet topology.
• More studies on the MLW model are still needed,
  for example, spectral analysis, fault tolerance
  and anti-attack properties ...

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Thank You.
Q A?