Generating Network Topologies That Obey Power Laws

1

• Generating Network Topologies That Obey Power
  Laws
• Christopher R. Palmer and
• J. Gregory Steffan
• School of Computer Science
• Carnegie Mellon University

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What is a Power Law?
y ßxa
Log
Log

• Faloutsos et al. define four power laws
• they found laws in multiple Internet graphs
• others found similar laws, also for the Web

? the Internet obeys power laws
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What is a Topology Generator?

• Artificial network generation algorithm
• often used to evaluate new network schemes
• Do artificial networks obey power laws?
• artificial networks may not be realistic
• conclusions could be inaccurate

? can we generate these topologies?
?does it matter?
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Outline

• ? Do existing generators obey power laws?
• Can we generate graphs that obey power laws?
• Do power law graphs impact results?
• Related work
• Conclusions

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Existing Topology Generators

• Waxman
• place nodes randomly in 2-space
• add edges with probability P(u,v)ae-d/(ßL)

• N-level hierarchical
• connect random graphs in an N-level hierarchy

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Power Laws 1 and 2

• PL 1 Out-degree vs. Rank
• compute the out-degree of all nodes
• sort in descending order
• PL 2 Frequency vs. Out-degree
• compute the out-degree of all nodes
• compute the frequency of each out-degree

? Internet graphs obey
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PL 1 Out-degree vs. Rank
2-Level ?0.81
Waxman ?0.80
? 2-Level and Waxman do not obey
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PL 2 Frequency vs. Out-degree
2-Level ?0.23
Waxman ?0.45
? 2-Level Waxman REALLY do not obey!
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Power Laws 3 and 4

• PL 3 Hopcounts
• number of pairs of nodes within i hops
• PL 4 Eigenvalues
• compute the largest 10 eigenvalues ?i

Avi ?ivi
? Internet graphs obey
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PL 3 Hopcounts
Waxman ?0.96
2-Level ?0.98
? 2-Level and Waxman obey
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PL 4 Eigenvalues
2-Level ?0.65
Waxman ?0.98
? 2-Level and Waxman obey
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Outline

• ? Do existing generators obey power laws?
• ? Can we generate graphs that obey power laws?
• Power-Law Out-Degree (PLOD)
• Recursive
• Do power law graphs impact results?
• Related work
• Conclusions

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Power-Law Out-Degree Algorithm (PLOD)

• FOR i1..N
• x uniform_random(1,N)
• out_degreei ßx-a
• FOR i1..M
• WHILE 1
• r uniform_random(1,N), c
  uniform_random(1,N)
• IF r ! c AND out_degreer AND out_degreec AND
  !Ar,c
• out_degreer--, out_degreec--
• Ar,c 1, Ac,r 1
• BREAK

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PLOD Example Topology
? 32 nodes, 48 links
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Recursive Topology Generator
80/20 Distribution
80
20
a
ß
Our Recursive Distribution
?
e
abge 1
? generalize to a 2D adjacency matrix
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Recursive Topology Generation
Link Probabilities
10 Generated links
? darker means higher probability / weight
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Recursive Topology Example
? 32 nodes, 50 low latency, 10 high latency (red)
links
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PL 1 Out-degree vs. Rank
Recursive ?0.89
PLOD ?0.97
? PLOD EXCELLENT power-law
? Recursive good power-law tail, non-power-law
start
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PL 2 Frequency vs. Degree
Recursive ?0.92
PLOD ?0.93
? both GOOD power-laws
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PL 3 Hopcounts
Recursive ?0.94
PLOD ?0.98
? both EXCELLENT power-laws
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PL 4 Eigenvalues
PLOD ?0.98
Recursive ?0.93
? both EXCELLENT power-laws
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Power-Law Summary Correlations
? GREEN cells obey power-laws, RED cells do not
? our generators have better Internet
characteristics!
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Outline

• ? Do existing generators obey power laws?
• ? Can we generate graphs that obey power laws?
• ? Do power law graphs impact results?
• Related work
• Conclusions

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STORM Multicast Algorithm

• ? client requests repair from parent with a nack

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Simulation Methodology

• Original STORM study
• used 2-level random topology
• source and clients connected to second-level
• Generating comparable topologies
• equalize graph size and average out-degree
• selection of high and low latency links
• What impact do we expect of PL topologies?
• average results will be similar
• distributions will differ

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STORM Average Overhead
? STORM overhead averages scale for all topologies
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STORM Overhead Distribution
2-Level
? overhead distribution varies significantly by
topology
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Loss Distribution
? loss distribution also varies significantly by
topology
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Related Work

• Barabási et al. (Notre Dame)
• BRITE (Boston University)
• What causes power laws in the Internet?
• incremental growth
• preferential connectivity

? BRITE uses these factors to generate graphs
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Conclusions

• Existing generators do not obey all power-laws
• Our two topology generators do
• PLOD use power-law to generate node degrees
• recursive use 80/20 law to generate links
• Do power-law topologies have any impact?
• maybe changed distributions for STORM
• maybe not averages unchanged for STORM

? moral simulate with different generators!
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Backup Slides
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Generating Comparable Topologies

• Equalize graph characteristics
• number of nodes
• average out-degree
• Ensure connectedness
• randomly connect disconnected components
• Assign high/low-latency links
• Recursive algorithm provides a distinction
• method for putting low-lat. links near clients