CS 268: Computer Networking

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1 CS 268: Computer Networking L-15 Network Topology 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 1

2 Outline Motivation/Background Power Laws Optimization Models Graph Generation 3 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 2

3 Internet topologies AT&T AT&T MCI SPRINT MCI SPRINT Router level Autonomous System (AS) level 5 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 3

4 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 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 4

Abilene Internet2 Backbone 9 Front Range GigaPoP Arizona St. Oregon GigaPoP Pacific Northwest GigaPoP AMES NGIX CENIC U. Arizona Intermountain GigaPoP Pacific Wave U.

5 Abilene Internet2 Backbone 9 Front Range GigaPoP Arizona St. Oregon GigaPoP Pacific Northwest GigaPoP AMES NGIX CENIC U. Arizona Intermountain GigaPoP Pacific Wave U. Hawaii WIDE UniNet TransPAC/APAN Abilene Backbone Physical Connectivity (as of December 16, 2003) Gbps Gbps Gbps Gbps Great Plains UNM Qwest Labs U. Memphis Denver Seattle Sunnyvale North Texas GigaPoP Texas Tech Texas GigaPoP UT Austin UT-SW Med Ctr. Iowa St. ESnet Los Angeles Kansas City Houston LaNet Tulane U. U. Louisville OneNet Northern Lights Indianapolis GEANT Atlanta SOX SFGP/ AMPATH Miss State GigaPoP Florida A&M U. So. Florida Merit Chicago New York Wash D.C. PSC DARPA BossNet UMD NGIX U. Florida Indiana GigaPoP OARNET NCSA MREN NCNI/MCNC SINet WiscREN WPI NYSERNet Rutgers U. MAGPI StarLight Northern Crossroads SURFNet MANLAN Mid-Atlantic Crossroads Drexel U. U. Delaware 10 5

6 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 11 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 6

7 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) Modeling Approach Random graph (Waxman88) Structural models (GT-ITM Calvert/Zegura, 1996) Degree-based models replicate power-law degree sequences Physical networks have hard technological (and economic) constraints. Optimization-driven models topologies consistent with design tradeoffs of network engineers 13 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 14 7

8 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 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 8

9 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 Outline Motivation/Background Power Laws Optimization Models Graph Generation 18 9

10 Power laws in AS level topology 19 Power Laws and Internet Topology Most nodes have few connections Source: Faloutsos et al. (1999) R(d) = P (D>d) x #nodes Rank R(d) Degree d A few nodes have lots of connections Router-level graph & Autonomous System (AS) graph Led to active research in degree-based network models 10

11 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 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)/σd(w) Connect degree 1 nodes using preferential connectivity Add remaining edges using preferential connectivity 22 11

12 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 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 existing node new node may contain multi-edges, self-loops 24 12

13 Features of Degree-Based Models Preferential Attachment Expected Degree Sequence 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 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 13

14 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 Outline Motivation/Background Power Laws Optimization Models Graph Generation 28 14

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

16 Aggregate Router Feasibility Total Router BW (Mbps) core technologies older/cheaper technologies approximate aggregate feasible region edge technologies degree cisco cisco cisco cisco cisco 7500 cisco 7200 linksys 4-port router ubr7246 cmts (cable) cisco 6260 dslam (DSL) cisco AS5850 (dialup) Source: Cisco Product Catalog, June Variability in End-User Bandwidths Connection Speed (Mbps) 1e4 1e3 1e2 1e1 1 1e-1 1e-2 Ethernet 1-10Gbps a few users have very high speed connections high performance computing academic and corporate Ethernet Mbps most users have low speed connections Broadband Cable/DSL ~500Kbps residential and small business Dial-up ~56Kbps 1 1e2 1e4 1e6 1e8 Rank (number of users) 32 16

17 Heuristically Optimal Topology Mesh-like core of fast, low degree routers Cores High degree nodes are at Edges the edges. Hosts 33 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 Step 2: Compute traffic demand B j x ij x B B ij i j Bandwidth (Mbps) Abstracted Technologically Feasible Region degree Step 3: Compute max flow max α s. t. x x ij i, j i, j ij i, j: k r ij B i = max B, k k αb B i j 34 17

18 Likelihood-Related Metric Define the metric L( g) = i, j connected d i d 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? j (d i = degree of node i) 35 HOT Abilene-inspired Sub-optimal P(g) Perfomance (bps) PA PLRG/GRG L max l(g) = 1 P(g) = 1.08 x l(g) = Relative Likelihood 36 18

19 Structure Determines Performance HOT PA PLRG/GRG P(g) = 1.13 x P(g) = 1.19 x P(g) = 1.64 x Achieved BW (Gbps) Achieved BW (Gbps) Achieved BW (Gbps) Degree Degree Degree 37 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 19

20 Outline Motivation/Background Power Laws Optimization Models Graph Generation 39 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 20

21 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 0K Average degree <k> 21

22 1K Degree distribution P(k) 2K Joint degree distribution P(k 1,k 2 ) 22

23 3K Joint edge degree distribution P(k 1,k 2,k 3 ) 3K, more exactly 23

24 4K Definition of dk-distributions dk-distributions are degree correlations within simple connected graphs of size d 24

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

25 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 (dkgraphs 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) Rewiring 50 25

26 Graph Reproduction

27 Power Laws Faloutsos 3 (Sigcomm 99) frequency vs. degree topology from BGP tables of 18 routers 53 Power Laws Faloutsos 3 (Sigcomm 99) frequency vs. degree topology from BGP tables of 18 routers 54 27

28 Power Laws Faloutsos 3 (Sigcomm 99) frequency vs. degree topology from BGP tables of 18 routers 55 Power Laws Faloutsos frequency vs. degree empirical ccdf P(d>x) ~ x -a 56 28

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

Today s Lecture : Computer Networking. Outline. Why study topology? Structural generators Power laws, HOT graphs,..

Today s Lecture : Computer Networking. Outline. Why study topology? Structural generators Power laws, HOT graphs,.. Today s Lecture 5-744: Computer Networking L-4 Network Topology Structural generators Power laws, HOT graphs,.. Assigned reading A First Principles Approach to Understanding the Internet s Router-level