1. 15-744: Computer Networking L-14 Network Topology

2. Sensor Networks Structural generators Power laws HOT graphs Graph generators Assigned reading On Power-Law Relationships of the Internet Topology A First Principles Approach to Understanding the Internet’s Router-level Topology 2

3. Outline Motivation/Background Power Laws Optimization Models Graph Generation 3

4. Why study topology? Correctness of network protocols typically independent of topology Performance of networks critically dependent on topology e.g., convergence of route information Internet impossible to replicate Modeling of topology needed to generate test topologies 4

5. Internet topologies AT&T SPRINT MCI AT&T MCISPRINT Router level Autonomous System (AS) level 5

6. More on topologies.. Router level topologies reflect physical connectivity between nodes Inferred from tools like traceroute or well known public measurement projects like Mercator and Skitter AS graph reflects a peering relationship between two providers/clients Inferred from inter-domain routers that run BGP and publlic projects like Oregon Route Views Inferring both is difficult, and often inaccurate 6

7. Hub-and-Spoke Topology Single hub node Common in enterprise networks Main location and satellite sites Simple design and trivial routing Problems Single point of failure Bandwidth limitations High delay between sites Costs to backhaul to hub 7

8. Simple Alternatives to Hub-and-Spoke Dual hub-and-spoke Higher reliability Higher cost Good building block Levels of hierarchy Reduce backhaul cost Aggregate the bandwidth Shorter site-to-site delay … 8

9. Abilene Internet2 Backbone 9

10. 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 10

11. Points-of-Presence (PoPs) Inter-PoP links Long distances High bandwidth Intra-PoP links Short cables between racks or floors Aggregated bandwidth Links to other networks Wide range of media and bandwidth Intra-PoP Other networks Inter-PoP 11

12. Deciding Where to Locate Nodes and Links Placing Points-of-Presence (PoPs) Large population of potential customers Other providers or exchange points Cost and availability of real-estate Mostly in major metropolitan areas Placing links between PoPs Already fiber in the ground Needed to limit propagation delay Needed to handle the traffic load 12

13. Trends in Topology Modeling Observation Long-range links are expensive Real networks are not random, but have obvious hierarchy Internet topologies exhibit power law degree distributions (Faloutsos et al., 1999) Physical networks have hard technological (and economic) constraints. Modeling Approach Random graph (Waxman88) Structural models (GT-ITM Calvert/Zegura, 1996) Degree-based models replicate power-law degree sequences Optimization-driven models topologies consistent with design tradeoffs of network engineers 13

14. Waxman model (Waxman 1988) Router level model Nodes placed at random in 2-d space with dimension L Probability of edge (u,v): ae^{-d/(bL)}, where d is Euclidean distance (u,v), a and b are constants Models locality 14 v u d(u,v)

15. Real world topologies Real networks exhibit Hierarchical structure Specialized nodes (transit, stub..) Connectivity requirements Redundancy Characteristics incorporated into the Georgia Tech Internetwork Topology Models (GT-ITM) simulator (E. Zegura, K.Calvert and M.J. Donahoo, 1995) 15

16. Transit-stub model (Zegura 1997) Router level model Transit domains placed in 2-d space populated with routers connected to each other Stub domains placed in 2-d space populated with routers connected to transit domains Models hierarchy 16

17. So…are we done? No! In 1999, Faloutsos, Faloutsos and Faloutsos published a paper, demonstrating power law relationships in Internet graphs Specifically, the node degree distribution exhibited power laws That Changed Everything….. 17

18. Outline Motivation/Background Power Laws Optimization Models Graph Generation 18

19. Power laws in AS level topology 19

20. 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

21. GT-ITM abandoned.. GT-ITM did not give power law degree graphs New topology generators and explanation for power law degrees were sought Focus of generators to match degree distribution of observed graph 21

22. Inet (Jin 2000) Generate degree sequence Build spanning tree over nodes with degree larger than 1, using preferential connectivity randomly select node u not in tree join u to existing node v with probability d(v)/  d(w) Connect degree 1 nodes using preferential connectivity Add remaining edges using preferential connectivity 22

23. Power law random graph (PLRG) Operations assign degrees to nodes drawn from power law distribution create kv copies of node v; kv degree of v. randomly match nodes in pool aggregate edges may be disconnected, contain multiple edges, self-loops contains unique giant component for right choice of parameters 23 2 11

24. Barabasi model: fixed exponent incremental growth initially, m0 nodes step: add new node i with m edges linear preferential attachment connect to node i with probability ki / ∑ kj 24 0.5 0.25 0.50.25 new nodeexisting node may contain multi-edges, self-loops

25. 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 25 Preferential AttachmentExpected Degree Sequence

26. Does Internet graph have these properties? No…(There is no Memphis!) Emphasis on degree distribution - structure ignored Real Internet very structured Evolution of graph is highly constrained 26

27. Problem With Power Law... but they're descriptive models! No correct physical explanation, need an understanding of: the driving force behind deployment the driving force behind growth 27

28. Outline Motivation/Background Power Laws Optimization Models Graph Generation 28

29. 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 29

30. 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 30 Cisco 12416 GSR, circa 2002 high BW low degree high degree low BW

31. approximate aggregate feasible region Aggregate Router Feasibility core technologies edge technologies older/cheaper technologies Source: Cisco Product Catalog, June 2002 31

32. 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 32

33. Heuristically Optimal Topology Hosts Edges Cores Mesh-like core of fast, low degree routers High degree nodes are at the edges. 33

34. Comparison Metric: 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 34

35. 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 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? Define the metric(d i = degree of node i) 35

36. 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 36

37. 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 37

38. Summary Network Topology Faloutsos 3 [SIGCOMM99] on Internet topology Observed many “power laws” in the Internet structure Router level connections, AS-level connections, neighborhood sizes Power law observation refuted later, Lakhina [INFOCOM00] Inspired many degree-based topology generators Compared properties of generated graphs with those of measured graphs to validate generator What is wrong with these topologies? Li et al [SIGCOMM04] Many graphs with similar distribution have different properties Random graph generation models don’t have network-intrinsic meaning Should look at fundamental trade-offs to understand topology Technology constraints and economic trade-offs Graphs arising out of such generation better explain topology and its properties, but are unlikely to be generated by random processes! 38

39. Outline Motivation/Background Power Laws Optimization Models Graph Generation 39

40. Graph Generation Many important topology metrics Spectrum Distance distribution Degree distribution Clustering… No way to reproduce most of the important metrics No guarantee there will not be any other/new metric found important 40

41. dK-series approach Look at inter-dependencies among topology characteristics See if by reproducing most basic, simple, but not necessarily practically relevant characteristics, we can also reproduce (capture) all other characteristics, including practically important Try to find the one(s) defining all others

42. 0K Average degree

43. 1K Degree distribution P(k)

44. 2K Joint degree distribution P(k 1,k 2 )

45. 3K “Joint edge degree” distribution P(k 1,k 2,k 3 )

46. 3K, more exactly

47. 4K

48. Definition of dK-distributions dK-distributions are degree correlations within simple connected graphs of size d

49. Nice properties of properties P d Constructability: we can construct graphs having properties P d (dK-graphs) Inclusion: if a graph has property P d, then it also has all properties P i, with i < d (dK- graphs are also iK-graphs) Convergence: the set of graphs having property P n consists only of one element, G itself (dK-graphs converge to G)

50. Rewiring 50

51. Graph Reproduction 51

52. Other Useful Tools Rocketfuel Biases? RouteViews 52

53. 53

54. Faloutsos 3 (Sigcomm’99) frequency vs. degree Power Laws topology from BGP tables of 18 routers 54

55. Faloutsos 3 (Sigcomm’99) frequency vs. degree Power Laws topology from BGP tables of 18 routers 55

56. Faloutsos 3 (Sigcomm’99) frequency vs. degree Power Laws topology from BGP tables of 18 routers 56

57. Faloutsos frequency vs. degree empirical ccdf P(d>x) ~ x -a Power Laws 57

58. Power Laws Faloutsos 3 (Sigcomm’99) frequency vs. degree empirical ccdf P(d>x) ~ x -a α ≈1.15 58