Topology Modeling: First-Principles Approach

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Topology Modeling First-Principles Approach

• Aditya Akella
• Supplemental Slides
• 03/30/2007

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Challenges
Why Topology Modeling

• Evaluate performance of protocols
• Protect Internet
• Resource provisioning
• Understand large scale networks

• Large Size
• Real topologies are not publicly available
• Incredible variability in many aspects

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Trends in Topology Modeling

• Observation

• Modeling Approach

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Power Laws and Internet Topology
Source Faloutsos et al. (1999)
Rank R(d)
R(d) P (Dgtd) x nodes
Degree d

• Router-level graph Autonomous System (AS) graph
• Led to active research in degree-based network
  models

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Degree-Based Models of Topology

• Preferential Attachment
• Growth by sequentially adding new nodes
• New nodes connect preferentially to nodes having
  more connections
• Examples Inet, GPL, AB, BA, BRITE, CMU power-law
  generator

• Expected Degree Sequence
• Based on random graph models that skew
  probability distribution to produce power laws in
  expectation
• Examples Power Law Random Graph (PLRG),
  Generalized Random Graph (GRG)

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Features of Degree-Based Models
Preferential Attachment
Expected Degree Sequence

• Degree sequence follows a power law (by
  construction)
• High-degree nodes correspond to highly connected
  central hubs, which are crucial to the system
• Achilles heel robust to random failure, fragile
  to specific attack

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Li et al.

• Consider the explicit design of the Internet
• Annotated network graphs (capacity, bandwidth)
• Technological and economic limitations
• Network performance
• Seek a theory for Internet topology that is
  explanatory and not merely descriptive.
• Explain high variability in network connectivity
• Ability to match large scale statistics (e.g.
  power laws) is only secondary evidence

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Router Technology Constraint
Cisco 12416 GSR, circa 2002
Total Bandwidth
Bandwidth per Degree
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Aggregate Router Feasibility
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Variability in End-User Bandwidths
1e4
Ethernet 1-10Gbps
1e3
1e2
Ethernet 10-100Mbps
Connection Speed (Mbps)
a few users have very high speed connections
1e1
Broadband Cable/DSL 500Kbps
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1e-1
Dial-up 56Kbps
most users have low speed connections
1e-2
1e6
1e2
1
1e4
1e8
Rank (number of users)
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Heuristically Optimal Topology
Mesh-like core of fast, low degree routers
Cores
High degree nodes are at the edges.
Edges
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Northern Lights
Merit
OneNet
Kansas City
Indian- apolis
Denver
Chicago
Seattle
New York
Wash D.C.
Sunnyvale
Los Angeles
Atlanta
Houston
PSC
Abilene Backbone Physical Connectivity (as of
December 16, 2003)
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Metrics for Comparison Network Performance

• Given realistic technology constraints on
  routers, how well is the network able to carry
  traffic?

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Structure Determines Performance
PA
PLRG/GRG
HOT
P(g) 1.19 x 1010
P(g) 1.64 x 1010
P(g) 1.13 x 1012
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Likelihood-Related Metric
Define the metric
(di degree of node i)

• Easily computed for any graph
• Depends on the structure of the graph, not the
  generation mechanism
• Measures how hub-like the network core is

For graphs resulting from probabilistic
construction (e.g. PLRG/GRG), LogLikelihood
(LLH) ? L(g) Interpretation How likely is a
particular graph (having given node degree
distribution) to be constructed?
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PA
PLRG/GRG
Abilene-inspired
Sub-optimal
HOT
P(g) Perfomance (bps)

Lmax l(g) 1 P(g) 1.08 x 1010