1 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Generating Network Topologies That Obey Power Laws Christopher R. Palmer.

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


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


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?


OutlineOutline

Do existing generators obey power laws?

• Can we generate graphs that obey power laws?

• Do power law graphs impact results?

• Related work

• Conclusions


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


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


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


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


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]


PL #3: HopcountsPL #3: Hopcounts

2-Level and Waxman obey

Waxman: ρ=0.96

2-Level: ρ=0.98


PL #4: EigenvaluesPL #4: Eigenvalues

2-Level and Waxman obey

Waxman: ρ=0.98

2-Level: ρ=0.65


OutlineOutline


Can we generate graphs that obey power laws?– Power-Law Out-Degree (PLOD) – Recursive

• Do power law graphs impact results?

• Related work

• Conclusions


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


PLOD: Example TopologyPLOD: Example Topology

32 nodes, 48 links


Recursive Topology GeneratorRecursive Topology Generator

β

γ

α Our Recursive Distribution:

80/20 Distribution: 80% 20%

generalize to a 2D adjacency matrix


Recursive Topology: GenerationRecursive Topology: Generation

Link Probabilities 10 Generated links

darker means higher probability / weight


Recursive Topology: ExampleRecursive Topology: Example

32 nodes, 50 low latency, 10 high latency (red) links


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


PL #2: Frequency vs. DegreePL #2: Frequency vs. Degree

both GOOD power-laws

Recursive: ρ=0.92

PLOD: ρ=0.93


PL #3: HopcountsPL #3: Hopcounts

both EXCELLENT power-laws

Recursive: ρ=0.94

PLOD: ρ=0.98


PL #4: EigenvaluesPL #4: Eigenvalues

both EXCELLENT power-laws

Recursive: ρ=0.93

PLOD: ρ=0.98


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!


OutlineOutline


Can we generate graphs that obey power laws?

Do power law graphs impact results?

• Related work

• Conclusions


STORM Multicast AlgorithmSTORM Multicast Algorithm

client requests repair from parent with a nack

source

client (parent)

clientnackrepair


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


STORM Average Overhead STORM Average Overhead

STORM overhead averages scale for all topologies


STORM Overhead DistributionSTORM Overhead Distribution

overhead distribution varies significantly by topology

2-Level


Loss DistributionLoss Distribution

loss distribution also varies significantly by topology


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


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!


Backup SlidesBackup Slides


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

1 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Generating Network Topologies That Obey Power Laws Christopher R. Palmer.

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