1 generating network topologies that obey power lawspalmer/steffan carnegie mellon generating...

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

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

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?

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

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

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

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

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

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]

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

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

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

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

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

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

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

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

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

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

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

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!

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

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

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

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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STORM Overhead DistributionSTORM Overhead Distribution

overhead distribution varies significantly by topology

2-Level

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Loss DistributionLoss Distribution

loss distribution also varies significantly by topology

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

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!

31Generating Network Topologies That Obey Power Laws Palmer/SteffanCarnegie Mellon

Backup SlidesBackup Slides

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