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  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

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

  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

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


  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

  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

  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 …

  9. Abilene Internet2 Backbone

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

  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 Inter-PoP Intra-PoP Other networks

  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

  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

  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 u d(u,v) 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)

  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

  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…..

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

  19. Power laws in AS level topology

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

  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

  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

  23. 2 1 1 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

  24. 0.5 0.5 0.25 0.5 0.25 existing node new node 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 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 Preferential Attachment Expected 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

  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

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

  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

  30. 3 10 high BW low degree high degree low BW 2 10 1 10 Bandwidth (Gbps) 15 x 10 GE 15 x 3 x 1 GE 0 10 15 x 4 x OC12 15 x 8 FE Technology constraint -1 10 0 1 2 Degree 10 10 10 Router Technology Constraint Cisco 12416 GSR, circa 2002 Total Bandwidth Bandwidth per Degree

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

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

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

  34. Given realistic technology constraints on routers, how well is the network able to carry traffic? Step 1: Constrain to be feasible Step 2: Compute traffic demand 1000000 100000 10000 Bj Abstracted Technologically Feasible Region 1000 Bandwidth (Mbps) 100 Step 3: Compute max flow xij 10 degree 1 10 100 1000 Bi Comparison Metric: Network Performance

  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 (di = degree of node i)

  36. 12 10 11 10 10 10 0 0.2 0.4 0.6 0.8 1 l(g) = Relative Likelihood PA Abilene-inspired Sub-optimal PLRG/GRG HOT P(g) Perfomance (bps) Lmax l(g) = 1 P(g) = 1.08 x 1010

  37. Structure Determines Performance HOT PA PLRG/GRG P(g) = 1.13 x 1012 P(g) = 1.19 x 1010 P(g) = 1.64 x 1010

  38. Summary Network Topology • Faloutsos3[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!

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

  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

  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 <k>

  43. 1K Degree distribution P(k)

  44. 2K Joint degree distribution P(k1,k2)

  45. 3K “Joint edge degree” distribution P(k1,k2,k3)

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

  50. Rewiring