15-744: Computer Networking

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15-744: Computer Networking

  1. 1. Sensor Networks • Structural generators • Power laws 15-744: Computer Networking • HOT graphs • Graph generators • Assigned reading • On Power-Law Relationships of the Internet L-14 Network Topology Topology • A First Principles Approach to Understanding the Internet’s Router-level Topology 2 Outline Why study topology? • Motivation/Background • Correctness of network protocols typically independent of topology • Power Laws • Performance of networks critically dependent on topology • e.g., convergence of route information • Optimization Models • Internet impossible to replicate • Modeling of topology needed to generate • Graph Generation test topologies 3 4 1
  2. 2. Internet topologies 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 AT&T AT&T • 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 MCI SPRINT • Inferring both is difficult, and often inaccurate MCI SPRINT Router level Autonomous System (AS) level 5 6 Hub-and-Spoke Topology Simple Alternatives to Hub-and-Spoke • Single hub node • Dual hub-and-spoke • Higher reliability • Common in enterprise networks • Higher cost • Main location and satellite sites • Good building block • Simple design and trivial routing • Problems • Levels of hierarchy • Single point of failure • Reduce backhaul cost • Aggregate the • Bandwidth limitations bandwidth • High delay between sites • Shorter site-to-site delay • Costs to backhaul to hub … 7 8 2
  3. 3. Intermountain U. Memphis Northern Lights GigaPoP Abilene Internet2 Backbone Front Range GigaPoP Great Plains U. Louisville Merit Indiana GigaPoP OARNET OneNet WiscREN Qwest Labs Arizona St. NCSA StarLight U. Arizona Iowa St. MREN Oregon NYSERNet GigaPoP Pacific Kansas UNM City Indian- WPI Wave Denver apolis Northern Pacific Crossroads Northwest Seattle Chicago GigaPoP SINet U. Hawaii ESnet New York SURFNet AMES NGIX GEANT Wash Rutgers U. WIDE Sunnyvale D.C. CENIC Los Angeles MANLAN UniNet Houston Atlanta North Texas GigaPoP SOX MAGPI TransPAC/APAN PSC Texas Tech SFGP/ DARPA Mid-Atlantic Abilene Backbone AMPATH BossNet Texas Crossroads Physical Connectivity GigaPoP (as of December 16, 2003) Miss State UMD NGIX Drexel U. GigaPoP 0.1-0.5 Gbps UT Austin 0.5-1.0 Gbps U. Florida LaNet Florida A&M U. Delaware 1.0-5.0 Gbps UT-SW 5.0-10.0 Gbps Med Ctr. NCNI/MCNC Tulane U. U. So. Florida 9 10 Points-of-Presence (PoPs) Deciding Where to Locate Nodes and Links • Inter-PoP links • Placing Points-of-Presence (PoPs) • Long distances Inter-PoP • Large population of potential customers • High bandwidth Intra-PoP • Other providers or exchange points • Intra-PoP links • Cost and availability of real-estate • Short cables between racks or floors • Mostly in major metropolitan areas • Aggregated bandwidth • Placing links between PoPs • Links to other • Already fiber in the ground networks Other networks • Needed to limit propagation delay • Wide range of media and bandwidth • Needed to handle the traffic load 11 12 3
  4. 4. Trends in Topology Modeling Waxman model (Waxman 1988) Observation Modeling Approach • Router level model • Long-range links are expensive • Random graph (Waxman88) • Nodes placed at random • Real networks are not random, • Structural models (GT-ITM in 2-d space with but have obvious hierarchy Calvert/Zegura, 1996) dimension L u d(u,v) • Degree-based models replicate • Internet topologies exhibit power-law degree sequences • Probability of edge (u,v): v power law degree distributions (Faloutsos et al., 1999) • ae^{-d/(bL)}, where d is Euclidean distance (u,v), a • Physical networks have hard • Optimization-driven models technological (and economic) topologies consistent with design and b are constants constraints. tradeoffs of network engineers • Models locality 13 14 Real world topologies Transit-stub model (Zegura 1997) • Real networks exhibit • Router level model • Hierarchical structure • Transit domains • placed in 2-d space • Specialized nodes (transit, stub..) • populated with routers • Connectivity requirements • connected to each other • Redundancy • Stub domains • placed in 2-d space • Characteristics incorporated into the • populated with routers Georgia Tech Internetwork Topology Models • connected to transit (GT-ITM) simulator (E. Zegura, K.Calvert domains and M.J. Donahoo, 1995) • Models hierarchy 15 16 4
  5. 5. So…are we done? Outline • No! • Motivation/Background • In 1999, Faloutsos, Faloutsos and Faloutsos published a paper, demonstrating • Power Laws power law relationships in Internet graphs • Specifically, the node degree distribution • Optimization Models exhibited power laws • Graph Generation That Changed Everything….. 17 18 Power laws in AS level topology Power Laws • Faloutsos3 (Sigcomm’99) • frequency vs. degree topology from BGP tables of 18 routers 19 20 5
  6. 6. Power Laws Power Laws • Faloutsos3 (Sigcomm’99) • Faloutsos3 (Sigcomm’99) • frequency vs. degree • frequency vs. degree topology from BGP tables of 18 routers topology from BGP tables of 18 routers 21 22 Power Laws Power Laws • Faloutsos • Faloutsos3 (Sigcomm’99) • frequency vs. • frequency vs. degree degree 1.15 • empirical ccdf • empirical ccdf P(d>x) ~ x-a P(d>x) ~ x-a 23 24 6
  7. 7. GT-ITM abandoned.. Inet (Jin 2000) • GT-ITM did not give power law degree • Generate degree sequence graphs • Build spanning tree over nodes with degree larger than 1, • New topology generators and explanation using preferential connectivity for power law degrees were sought • randomly select node u not in • Focus of generators to match degree tree distribution of observed graph • 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 25 26 Power law random graph (PLRG) Barabasi model: fixed exponent • Operations • incremental growth • assign degrees to nodes drawn from power law distribution • create kv copies of node v; kv degree of v. • initially, m0 nodes • randomly match nodes in pool • step: add new node i with m edges • aggregate edges • linear preferential attachment 2 • connect to node i with probability (ki) = ki / kj 0.5 0.5 0.25 1 1 may be disconnected, contain multiple edges, self-loops 0.5 0.25 • contains unique giant component for right choice of existing node new node parameters may contain multi-edges, self-loops 27 28 7
  8. 8. Features of Degree-Based Models Does Internet graph have these properties? Preferential Attachment Expected Degree Sequence • No…(There is no Memphis!) • Emphasis on degree distribution - structure ignored • Real Internet very structured • Evolution of graph is highly constrained • 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 29 30 Problem With Power Law Outline • Motivation/Background • ... but they're descriptive models! • Power Laws • No correct physical explanation, need an understanding of: • the driving force behind deployment • Optimization Models • the driving force behind growth • Graph Generation 31 32 8
  9. 9. Li et al. Router Technology Constraint 3 10 • Consider the explicit design of the Internet Cisco 12416 GSR, circa 2002 high BW low • Annotated network graphs (capacity, degree Total Bandwidth high degree bandwidth) 10 2 low BW Bandwidth (Gbps) • Technological and economic limitations • Network performance 1 Bandwidth per Degree 10 • Seek a theory for Internet topology that is explanatory and not merely descriptive. 15 x 10 GE 15 x 3 x 1 GE 0 10 • Explain high variability in network connectivity 15 x 4 x OC12 15 x 8 FE • Ability to match large scale statistics (e.g. Technology constraint power laws) is only secondary evidence 10 -1 0 1 2 10 10 10 33 Degree 34 Aggregate Router Feasibility Variability in End-User Bandwidths core technologies 1e4 high approximate aggregate Ethernet performance feasible region 1-10Gbps 1e3 computing academic Connection Speed (Mbps) and corporate 1e2 a few users have very Ethernet residential and high speed 10-100Mbps 1e1 connections small business older/cheaper technologies 1 Broadband edge technologies Cable/DSL ~500Kbps 1e-1 most users Dial-up have low speed ~56Kbps connections 1e-2 Source: Cisco Product Catalog, June 2002 1 1e2 1e4 1e6 1e8 35 Rank (number of users) 36 9
  10. 10. Heuristically Optimal Topology Comparison Metric: Network Performance Given realistic technology constraints on routers, how well Mesh-like core of fast, low degree routers is the network able to carry traffic? Step 1: Constrain to Step 2: Compute traffic demand be feasible Bj 1000000 100000 xij Bandwidth (Mbps) 10000 Bi 1000 Abstracted High degree nodes Step 3: Compute max flow 100 Technologically are at the edges. Feasible Region 10 degree 1 10 100 1000 37 38 Likelihood-Related Metric Define the metric (di = degree of HOT Abilene-inspired Sub-optimal PA PLRG/GRG node i) P(g) Perfomance (bps) 12 • Easily computed for any graph 10 • Depends on the structure of the graph, not the generation mechanism • Measures how “hub-like” the network core is 11 • For graphs resulting from probabilistic construction (e.g. PLRG/ 10 GRG), LogLikelihood (LLH) L(g) Lmax • Interpretation: How likely is a particular graph (having given l(g) = 1 node degree distribution) to be constructed? 10 10 P(g) = 1.08 x 1010 0 0.2 0.4 0.6 0.8 1 39 l(g) = Relative Likelihood 40 10
  11. 11. Structure Determines Performance Summary Network Topology HOT PA PLRG/GRG • 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] P(g) = 1.13 x 1012 P(g) = 1.19 x 1010 P(g) = 1.64 x 1010 • 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 generted by random processes! 41 42 Outline Graph Generation • Motivation/Background • Many important topology metrics • Spectrum • Power Laws • Distance distribution • Degree distribution • Clustering… • Optimization Models • No way to reproduce most of the important metrics • Graph Generation • No guarantee there will not be any other/ new metric found important 43 44 11
  12. 12. dK-series approach 0K • Look at inter-dependencies among topology Average degree <k> 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 1K 2K Degree distribution P(k) Joint degree distribution P(k1,k2) 12
  13. 13. 3K 3K, more exactly “Joint edge degree” distribution P(k1,k2,k3) 4K Definition of dK-distributions dK-distributions are degree correlations within simple connected graphs of size d 13
  14. 14. Nice properties of properties Pd Rewiring • 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) 54 14

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