1. Topology Modeling: First-Principles Approach Aditya Akella Supplemental Slides 03/30/2007

2. Evaluate performance of protocols Protect Internet Resource provisioning Understand large scale networks Why Topology Modeling Challenges Large Size Real topologies are not publicly available Incredible variability in many aspects

3. Trends in Topology Modeling Observation Modeling Approach Real networks are not random, but have obvious hierarchy. Structural models (GT-ITM Calvert/Zegura, 1996) Long-range links are expensive Random graph models (Waxman, 1988) Internet topologies exhibit power law degree distributions (Faloutsos et al., 1999) Degree-based models replicate power-law degree sequences

4. A few nodes have lots of connections Rank R(d) Degree d Source: Faloutsos et al. (1999) Power Laws and Internet Topology Router-level graph & Autonomous System (AS) graph Led to active research in degree-based network models Most nodes have few connections R(d) = P (D>d) x #nodes

5. Degree-Based Models of Topology Expected Degree Sequence –Based on random graph models that skew probability distribution to produce power laws in expectation –Examples: Power Law Random Graph (PLRG), Generalized Random Graph (GRG) Preferential Attachment –Growth by sequentially adding new nodes –New nodes connect preferentially to nodes having more connections –Examples: Inet, GPL, AB, BA, BRITE, CMU power-law generator

6. Features of Degree-Based Models Degree sequence follows a power law (by construction) High-degree nodes correspond to highly connected central “ hubs ”, which are crucial to the system Achilles ’ heel: robust to random failure, fragile to specific attack Preferential AttachmentExpected Degree Sequence

7. Li et al. Consider the explicit design of the Internet –Annotated network graphs (capacity, bandwidth) –Technological and economic limitations –Network performance Seek a theory for Internet topology that is explanatory and not merely descriptive. –Explain high variability in network connectivity –Ability to match large scale statistics (e.g. power laws) is only secondary evidence

8. 10 0 1 2 Degree 10 10 0 1 2 3 Bandwidth (Gbps) 15 x 10 GE 15 x 3 x 1 GE 15 x 4 x OC12 15 x 8 FE Technology constraint Total Bandwidth Bandwidth per Degree Router Technology Constraint Cisco 12416 GSR, circa 2002 high BW low degree high degree low BW

9. approximate aggregate feasible region Aggregate Router Feasibility core technologies edge technologies older/cheaper technologies

10. Rank (number of users) Connection Speed (Mbps) 1e-1 1e-2 1 1e1 1e2 1e3 1e4 1e211e4 1e6 1e8 Dial-up ~56Kbps Broadband Cable/DSL ~500Kbps Ethernet 10-100Mbps Ethernet 1-10Gbps most users have low speed connections a few users have very high speed connections high performance computing academic and corporate residential and small business Variability in End-User Bandwidths

11. Heuristically Optimal Topology Hosts Edges Cores Mesh-like core of fast, low degree routers High degree nodes are at the edges.

12. SOX SFGP/ AMPATH U. Florida U. So. Florida Miss State GigaPoP WiscREN SURFNet Rutgers U. MANLAN Northern Crossroads Mid-Atlantic Crossroads Drexel U.U. Delaware PSC NCNI/MCNCMAGPI UMD NGIX DARPA BossNet GEANT Seattle Sunnyvale Los Angeles Houston Denver Kansas City Indian- apolis Atlanta Wash D.C. Chicago New York OARNET Northern Lights Indiana GigaPoP Merit U. Louisville NYSERNet U. Memphis Great Plains OneNet Arizona St. U. Arizona Qwest LabsUNM Oregon GigaPoP Front Range GigaPoP Texas TechTulane U. North Texas GigaPoP Texas GigaPoP LaNet UT Austin CENICUniNet WIDE AMES NGIX Pacific Northwest GigaPoP U. Hawaii Pacific Wave ESnet TransPAC/APAN Iowa St. Florida A&M UT-SW Med Ctr. NCSA MREN SINet WPI StarLight Intermountain GigaPoP Abilene Backbone Physical Connectivity (as of December 16, 2003) 0.1-0.5 Gbps 0.5-1.0 Gbps 1.0-5.0 Gbps 5.0-10.0 Gbps

13. Metrics for Comparison: Network Performance Given realistic technology constraints on routers, how well is the network able to carry traffic? Step 1: Constrain to be feasible Abstracted Technologically Feasible Region 1 10 100 1000 10000 100000 1000000 101001000 degree Bandwidth (Mbps) Step 3: Compute max flow BiBi BjBj x ij Step 2: Compute traffic demand

14. PAPLRG/GRGHOT Structure Determines Performance P(g) = 1.19 x 10 10 P(g) = 1.64 x 10 10 P(g) = 1.13 x 10 12

15. Likelihood-Related Metric Easily computed for any graph Depends on the structure of the graph, not the generation mechanism Measures how “hub-like” the network core is Define the metric( d i = degree of node i ) For graphs resulting from probabilistic construction (e.g. PLRG/GRG), LogLikelihood (LLH)  L(g) Interpretation: How likely is a particular graph (having given node degree distribution) to be constructed?

16. L max l(g) = 1 P(g) = 1.08 x 10 10 P(g) Perfomance (bps) PA PLRG/GRG HOT Abilene-inspiredSub-optimal 00.20.40.60.81 10 11 10 12 l(g) = Relative Likelihood