1 generating network topologies that obey power lawspalmer/steffan carnegie mellon generating...
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1Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
Generating Network Topologies That Obey Power Laws
Christopher R. Palmer and J. Gregory Steffan
School of Computer ScienceCarnegie Mellon University
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2Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
What is a Power Law?What is a Power Law?
Faloutsos et al. define four power laws:– they found laws in multiple Internet graphs
– others found similar laws, also for the Web
y = βxα
Log
Log
the Internet obeys power laws
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3Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
What is a Topology Generator?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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4Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
OutlineOutline
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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5Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
Existing Topology GeneratorsExisting Topology Generators
Waxman:– place nodes randomly in 2-space– add edges with probability P(u,v)=αe-d/(βL)
N-level hierarchical:–connect random graphs in an N-level hierarchy
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6Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
Power Laws 1 and 2Power 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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7Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
PL #1: Out-degree vs. RankPL #1: Out-degree vs. Rank
2-Level and Waxman do not obey
Waxman: ρ=0.80
2-Level: ρ=0.81
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8Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
PL #2: Frequency vs. Out-degreePL #2: Frequency vs. Out-degree
2-Level & Waxman REALLY do not obey!
Waxman: ρ=0.45
2-Level: ρ=0.23
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9Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
Power Laws 3 and 4Power Laws 3 and 4
PL #3: Hopcounts – number of pairs of nodes within i hops
PL #4: Eigenvalues– compute the largest 10 eigenvalues λi
Internet graphs obey
[A][vi] = λi[vi]
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10Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
PL #3: HopcountsPL #3: Hopcounts
2-Level and Waxman obey
Waxman: ρ=0.96
2-Level: ρ=0.98
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11Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
PL #4: EigenvaluesPL #4: Eigenvalues
2-Level and Waxman obey
Waxman: ρ=0.98
2-Level: ρ=0.65
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12Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
OutlineOutline
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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13Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
Power-Law Out-Degree Algorithm (PLOD)Power-Law Out-Degree Algorithm (PLOD)
FOR i:1..Nx = uniform_random(1,N)
out_degreei = βx-α
FOR i:1..MWHILE 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 = 1BREAK
Assign exponentialout-degree credits
Place an edge inthe adjacency matrix
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PLOD: Example TopologyPLOD: Example Topology
32 nodes, 48 links
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15Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
Recursive Topology GeneratorRecursive Topology Generator
β
γ
α Our Recursive Distribution:
80/20 Distribution: 80% 20%
generalize to a 2D adjacency matrix
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16Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
Recursive Topology: GenerationRecursive Topology: Generation
Link Probabilities 10 Generated links
darker means higher probability / weight
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17Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
Recursive Topology: ExampleRecursive Topology: Example
32 nodes, 50 low latency, 10 high latency (red) links
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18Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
PL #1: Out-degree vs. RankPL #1: Out-degree vs. Rank
Recursive: good power-law tail, non-power-law start
PLOD: EXCELLENT power-law
Recursive: ρ=0.89
PLOD: ρ=0.97
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19Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
PL #2: Frequency vs. DegreePL #2: Frequency vs. Degree
both GOOD power-laws
Recursive: ρ=0.92
PLOD: ρ=0.93
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20Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
PL #3: HopcountsPL #3: Hopcounts
both EXCELLENT power-laws
Recursive: ρ=0.94
PLOD: ρ=0.98
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21Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
PL #4: EigenvaluesPL #4: Eigenvalues
both EXCELLENT power-laws
Recursive: ρ=0.93
PLOD: ρ=0.98
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22Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
Power-Law Summary: CorrelationsPower-Law Summary: Correlations
PL #1:
Degree
PL #2:
Deg. Freq
PL #3:
Hops
PL #4:
Eigenval
2-Level .81 .23 .98 .65
Waxman .80 .45 .96 .97
PLOD .99 .93 .98 .98
Recursive .89 .92 .94 .93
GREEN cells obey power-laws, RED cells do not
our generators have better Internet characteristics!
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23Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
OutlineOutline
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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24Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
STORM Multicast AlgorithmSTORM Multicast Algorithm
client requests repair from parent with a nack
source
client (parent)
clientnackrepair
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25Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
Simulation MethodologySimulation 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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26Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
STORM Average Overhead STORM Average Overhead
STORM overhead averages scale for all topologies
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27Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
STORM Overhead DistributionSTORM Overhead Distribution
overhead distribution varies significantly by topology
2-Level
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28Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
Loss DistributionLoss Distribution
loss distribution also varies significantly by topology
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29Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
Related WorkRelated 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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30Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
ConclusionsConclusions
• 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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31Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
Backup SlidesBackup Slides
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32Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon
Generating Comparable TopologiesGenerating 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