1 / 32

Generating Network Topologies That Obey Power Laws Christopher R. Palmer and J. Gregory Steffan

Generating Network Topologies That Obey Power Laws Christopher R. Palmer and J. Gregory Steffan School of Computer Science Carnegie Mellon University. What is a Power Law?. Faloutsos et al . define four power laws: they found laws in multiple Internet graphs

derek-mckinney
Download Presentation

Generating Network Topologies That Obey Power Laws Christopher R. Palmer and J. Gregory Steffan

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. 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

  3. 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?

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

  5. Existing 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

  6. 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

  7. PL #1: Out-degree vs. Rank 2-Level: ρ=0.81 Waxman: ρ=0.80 2-Level and Waxman do not obey

  8. PL #2: Frequency vs. Out-degree 2-Level: ρ=0.23 Waxman: ρ=0.45 2-Level & Waxman REALLY do not obey!

  9. Power Laws 3 and 4 PL #3: Hopcounts • number of pairs of nodes within i hops PL #4: Eigenvalues • compute the largest 10 eigenvalues λi [A][vi] = λi[vi] Internet graphs obey

  10. PL #3: Hopcounts Waxman: ρ=0.96 2-Level: ρ=0.98 2-Level and Waxman obey

  11. PL #4: Eigenvalues 2-Level: ρ=0.65 Waxman: ρ=0.98 2-Level and Waxman obey

  12. 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

  13. Assign exponential out-degree credits Place an edge in the adjacency matrix Power-Law Out-Degree Algorithm (PLOD) FOR i:1..N x = uniform_random(1,N) out_degreei = βx-α FOR i:1..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

  14. PLOD: Example Topology 32 nodes, 48 links

  15. Recursive Topology Generator 80/20 Distribution: 80% 20% α β Our Recursive Distribution: γ e a+b+g+e = 1 generalize to a 2D adjacency matrix

  16. Recursive Topology: Generation Link Probabilities 10 Generated links darker means higher probability / weight

  17. Recursive Topology: Example 32 nodes, 50 low latency, 10 high latency (red) links

  18. PL #1: Out-degree vs. Rank Recursive: ρ=0.89 PLOD: ρ=0.97 PLOD: EXCELLENT power-law Recursive: good power-law tail, non-power-law start

  19. PL #2: Frequency vs. Degree Recursive: ρ=0.92 PLOD: ρ=0.93 both GOOD power-laws

  20. PL #3: Hopcounts Recursive: ρ=0.94 PLOD: ρ=0.98 both EXCELLENT power-laws

  21. PL #4: Eigenvalues PLOD: ρ=0.98 Recursive: ρ=0.93  both EXCELLENT power-laws

  22. Power-Law Summary: Correlations GREEN cells obey power-laws, RED cells do not our generators have better Internet characteristics!

  23. Outline  Do existing generators obey power laws?  Can we generate graphs that obey power laws?  Do power law graphs impact results? • Related work • Conclusions

  24. source  client nack repair client (parent) STORM Multicast Algorithm client requests repair from parent with a nack

  25. 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

  26. STORM Average Overhead STORM overhead averages scale for all topologies

  27. STORM Overhead Distribution 2-Level overhead distribution varies significantly by topology

  28. Loss Distribution loss distribution also varies significantly by topology

  29. 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

  30. 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!

  31. Backup Slides

  32. 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

More Related