MEDUSA

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MEDUSA New Model of Internet Topology Using
k-shell Decomposition

• Shai Carmi Shlomo Havlin

Bloomington 05/24/2005
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Who we are

• Talk prepared by Shai Carmi.
• Graduate student in the Department of Physics,
  Bar-Ilan University, Israel.
• Supervised by Prof. Shlomo Havlin, who gives
  the talk.

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Who we are

• Collaborators
• Prof. Scott Kirkpatrick, Hebrew University of
  Jerusalem, Israel.
• Dr. Yuval Shavitt, Tel-Aviv University, Israel.
• Eran Shir, Ph.D. Student, Tel-Aviv University,
  Israel.

Scott
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Measuring the Internet

• Previous efforts to measure the Internet have
  used
• One machine Traceroute to many destinations.
• Many machines, specially deployed to traceroute
  to many destinations
• Sites restricted to academic or govt labs, on
  network backbone
• General perception was that Law of Diminishing
  Returns has set in.

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Measuring the Internet

• DIMES Distributed Internet MEasurement and
  Simulations (http//www.netdimes.org), seems to
  have made a breakthrough.
• Dont manage machines, offer a very lightweight,
  limited purpose client, and collect its
  measurements centrally.
• 100 1000 clients via word-of-mouth (Sep04 to
  Apr05). gt5000 clients now, achieved via Science
  article, slashdot. 82 countries represented. 2-3
  M measurements per day.

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The network we analyze

• We consider the Internet at the level of its
  autonomous systems (ASes), with roughly 20,000
  nodes and 70,000 links.
• We use data gathered between March and June 2005.
  In the future, can study network dynamics, using
  intervals of months, weeks or even days.

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k-shell method

• Use recursive pruning to peel network layers.
• To remove the 1-shell, keep removing all nodes
  with one link (degree1) until only nodes with
  degree 2 or more remain.
• To remove the 2-shell, keep removing nodes with 2
  links, until all degrees are gt 3.
• Keep going until all nodes are removed.

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k-shell method - example
Original Graph
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k-shell method - example
Pruning Degree 1
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k-shell method - example
Keep Pruning Degree 1
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k-shell method - example
Keep Pruning Degree 1
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k-shell method - example
Pruning Degree 2
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k-shell method - example
Keep Pruning Degree 2
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k-shell method - example
Pruning Degree 3
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k-shell method

• Definitions
• k-Core union of all shells with indices gt k.
• k-Crust union of all shells withindices lt k.

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Applications

• Can use k-shell method to analyze the AS network.
• For example, color each node by its shell index
  to visualize the network.
• Next, plot quantities as a function of the shell
  index.
• Gain understanding of the network structure.
• More useful indicator then the degree.

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AS graph colored by shells
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Identification of a nucleus

• k-shell method enables us to identify the heart,
  or nucleus of the network as nodes in the last
  core.
• No parameters need to be fixed. (Topology
  dependent only).
• Stable over time.
• Significant ASes (tier-1) were verified to be in
  the nucleus.
• Most quantities show singular behavior at the
  last shell. Some examples -

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Number of nodes and degrees in the shells
Slope 2.6
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Centrality vs. shell
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Where links go
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Distances (vs. crusts)
Distances measured between all pairs in the
largest cluster of the crust
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Number of site-distinct paths in the nucleus
At least 41 distinct paths between each pair
41 is the k-shell index of the nucleus
The nucleus is k-connected!
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Beyond the nucleus

• It is left to understand the role of the other
  nodes in the network.
• We look at the connectivity properties of the
  crusts.
• Incorporate this with observations from
  previously shown plots.

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Clusters in the crusts
Percolation Threshold
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Structure of the AS network

• Nodes outside of the nucleus can be categorized
  into
• The fractal part nodes in the largest cluster
  of the one-before-last crust contains 70 of
  the nodes in the network.
• The rest of the nodes become then the isolated
  part.

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Properties of the fractal part

• Connected (by construction), so that routing is
  possible without traversing and congesting the
  nucleus.
• Connections to the nucleus decrease path lengths
  significantly.
• Show fractal properties and power-laws.

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Properties of the fractal part

• Fractal dimension calculated using the box cover
  method (SHM 2005).
• Crossover behavior between non-fractal and
  completely fractal at the percolation critical
  point.
• Percolation theory arguments predict

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Properties of the fractal part
Percolation theory prediction slope 2.5
The 6-crust is renormalized with box of size 4
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Properties of the isolated part

• Contains 30 of the ASes, not reachable without
  the nucleus.
• Low degree nodes, high clustering.
• Many small clusters.
• Contributions found in up to k10 shell.
• Many nodes are connected directly to highly
  connected nodes in the nucleus.

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AS network model

• We summarize the AS network is composed of 3
  main sub-components
• Nucleus Nodes in last shell.
• Fractal Part Nodes in the largest cluster of
  the one-before-last crust.
• Isolated Part Nodes in all-but-largest clusters
  of the one-before-last crust.
• We name this model Medusa because of its
  jellyfish like structure.
• Some similarities to Faloutsous Jellyfish model
  but important differences.

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Medusa model of the AS network
Our view of the Internet
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The End.Thank you for your attention.
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Comments

• Some properties (such as percolation) are found
  in the Random-Scale-Free-Model
• Internet might not be so special.
• To have more insight must investigate navigation
  with commercial restrictions Many properties
  change.

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Clustering coefficient vs. shell
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Nearest neighbor degree vs. shell
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Derivation of the fractal dimension

• At the threshold, almost all the high degree
  nodes are removed, such that the network becomes
  similar to a random (Erdos-Renyi) network.
• Percolation in random networks is equivalent to
  percolation in an infinite dimensional lattice,
  in which we know the fractal dimension of the
  largest component is 4, .
• For infinite dimensional lattices, .
• Thus we conclude, ,or the
  ''shortest-path'' fractal dimension is 2.

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Faloutsos Internet jellyfish model

• The Jellyfish Model
• Identify core of network as maximal clique. (?
  not a very robust or reproducible approach)
• Shells around network labeled by hop count from
  core (a small world)
• Find large portion of peripheral sites connect to
  core.

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Faloutsos Internet jellyfish model

• Faloutsos view of the internet