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Modeling Internet Topology

Modeling Internet Topology. Ellen W. Zegura College of Computing Georgia Tech. Outline. Part I - Modeling topology Background Survey of models + what is known about topology Example: mathematical foundations of degree-based generation Evaluation of topologies Part II - Reality check

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Modeling Internet Topology

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  1. Modeling Internet Topology Ellen W. Zegura College of Computing Georgia Tech

  2. Outline • Part I - Modeling topology • Background • Survey of models + what is known about topology • Example: mathematical foundations of degree-based generation • Evaluation of topologies • Part II - Reality check • Beyond simple topology • Visualization • Open questions/Bold statements/Random thoughts • Reading list IPAM Workshop Tutorial

  3. Networking background transit domains domains/autonomous systems exchange point border routers peering hosts/endsystems routers stub domains lowly worm access networks IPAM Workshop Tutorial

  4. Topology modeling • Graph representation • Router-level modeling • vertices are routers • edges are one-hop IP connectivity • Domain- (AS-) level modeling • vertices are domains (ASes) • edges are peering relationships IPAM Workshop Tutorial

  5. Survey of models • Waxman (Waxman 1988) • router level model capturing locality • Transit-stub (Zegura 1997), Tiers (Doar 1997) • router level model capturing hierarchy • Inet (Jin 2000) • AS level model based on degree sequence • BRITE (Medina 2000) • AS level model based on evolution IPAM Workshop Tutorial

  6. 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 IPAM Workshop Tutorial

  7. 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 IPAM Workshop Tutorial

  8. Real data: AS topology • Oregon route view server; peers with routers to collect BGP routing tables • Data publicly available from Nov 97 to present (nlanr.org, routeviews.org) • Faloutsos 1999 • degree sequence approximated by power law • i.e., let f(d) be fraction of nodes with degree d, then f(d)  d^ • Chen 2002 • Oregon data incomplete (but so is theirs!) • degree sequence highly variable but not strict power law IPAM Workshop Tutorial

  9. 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 IPAM Workshop Tutorial

  10. BRITE (Medina 2000) • Generate small backbone, with nodes placed: • randomly or • concentrated (skewed) • Add nodes one at a time (incremental growth) • New node has constant # of edges connected using: • preferential connectivity and/or • locality IPAM Workshop Tutorial

  11. Router-level measurement source 0 • General technique: traceroute, returns list of IP addresses on a path from source to destination • Collection challenges: • obtaining sufficient traceroute origin points • deciding set of destination IP addresses (for coverage) • limiting traceroute load • Postprocessing challenges: • resolving aliases (which IP addresses belong to same router) S1 D1 destination 0 IPAM Workshop Tutorial

  12. Projects • Lucent (Burch 1999) • single source (Lucent), ~100k destinations • emphasis: longitudinal study, visualization • Skitter (Broido 2001) • 20 sources (“monitors”), ~400k destinations • emphasis: measurement repository, analysis • Mercator (Govindan 2000) • single source (but uses source routing), 150k interfaces • emphasis: heuristics for map construction IPAM Workshop Tutorial

  13. What is known? (hard to say) • Caveat: router-level mapping clearly incomplete, so conclusions are weak • Observations: • qualitatively similar to AS graph on a number of measures • Weibull distributions good fit for number of quantities (including degree distribution) IPAM Workshop Tutorial

  14. Outline • Part I - Modeling topology • Background • Survey of models + what is known about topology • Example: mathematical foundations • Evaluation of topologies • Part II - Reality check • Beyond simple topology • Visualization • Open questions/Bold statements/Random thoughts • Reading list IPAM Workshop Tutorial

  15. Foundations of degree-based generation (Mihail 2002) • Given degree sequence d(1) >= d(2) >= … >= d(n) • A degree sequence is realizable if there is a simple graph (no self-loops or multiple links) with this sequence • Necessary and sufficient condition for degree sequence to be realizable: • for each subset of k highest degree nodes, degrees can be “absorbed” within the nodes and the outside degrees IPAM Workshop Tutorial

  16. Construction algorithm • Maintain residual degrees of vertices, d(v) • Repeat until all vertices have been chosen: • pick arbitrary vertex v • add edges from v to d(v) vertices of highest residual degree • update residual degrees • Note: order to pick varbitrary IPAM Workshop Tutorial

  17. Sparse/dense core • Dense core • pick v’s starting with high degree vertices • will tend to connect high degree vertices • Sparse core • pick v’s starting with low degree vertices • less likely to connect high degree vertices IPAM Workshop Tutorial

  18. Example • Large topology (11000+ nodes, 32000+ edges) • Dense core • diameter 5 • average path length 3.6 • Sparse core • diameter 29 • average path length 17.9 IPAM Workshop Tutorial

  19. Random instance • Start from any realization of degree sequence • Pick two edges at random, (u,v) and (s,t), with distinct endpoints • If doesn’t disconnect graph, remove edges and insert (u,s) and (v,t) • Result satisfies degree sequence • In the limit, reaches every possible connected realization with equal probability u s v t u s v t IPAM Workshop Tutorial

  20. Example • Different starting points • Snapshots, 25k, 50k, 100k, 300k, 600k iters • Large topology, sparse initial core • diameter: 29, 13, 11, 11, 10, 10 • avgspl: 5.6, 3.6, 3.4, 3.4, 3.4, 3.4 • Large topology, dense initial core • diameter: 5, 10, 10, 10, 10, 10 • avgspl: 3.6, 3.2, 3.2, 3.4, 3.4, 3.4 IPAM Workshop Tutorial

  21. Notes about models • Variants on evolutionary models • Variants on degree-driven models • Appeal of evolutionary • Relationship to work on “networks” in general IPAM Workshop Tutorial

  22. Outline • Part I - Modeling topology • Background • Survey of models + what is known about topology • Example: mathematical foundations • Evaluation of topologies • Part II - Reality check • Beyond simple topology • Visualization • Open questions/Bold statements/Random thoughts • Reading list IPAM Workshop Tutorial

  23. Evaluation • Question: what determines whether a topology generator is “good”? • Essentially an unsolved (and hard) problem • depends on what topologies are used for • NOT “degree sequence follows a power law!” IPAM Workshop Tutorial

  24. Metrics • Path-related metrics • diameter, shortest path length • Clustering metrics • neighborhood size (“expansion”), eigenvalue decomposition, clustering coefficient • Robustness metrics • resilience • Hierarchy metrics • link usage, size of layers IPAM Workshop Tutorial

  25. Small world topologies (Bu 2002) • Defined by two measures: • characteristic path length L = number of edges in shortest path between two vertices, averaged over all vertex pairs • clustering coefficient C: • take vertex v with k  1 neighbors • at most k(k-1)/2 edges among neighbors • C(v) = fraction of k(k-1)/2 edges present • C = average clustering coefficient • C >> C_random, L  L_random k nodes v IPAM Workshop Tutorial

  26. Findings • AS-level topologies satisfy small-world test • Example Mar 00: • L=3.7, L_random=3.8 • C=.39, C_random=.0023 • Example, Sept 01: • L= 3.6, L_random=3.6 • C=.47, C_random=.0015 IPAM Workshop Tutorial

  27. Distinguishing between types of generators (Tangmunarunkit 2001) • Goal: large-scale metrics that distinguish between classes of graphs • Proposal: Expansion, resilience and distortion • differentiate between canonical graphs (mesh, tree, random graph) • differentiate between three types of generators • random graph (e.g., Waxman) • structural (e.g., Transit-Stub, Tiers) • degree-based (e.g., PLRG, BRITE) IPAM Workshop Tutorial

  28. Model “signatures” • Signature: expansion, resilience, distortion • Waxman: H H H (like random) • Tiers: L H L • Transit-stub: H L L (like tree) • PLRG: H H L (like complete graph) • Also: real topologies and other degree-based generators have H H L signature IPAM Workshop Tutorial

  29. Measure of hierarchy • link-value measure • see paper for details… • bottom line: degree-based generators contain loose notion of hierarchy that is somewhat similar to loose notion in Internet IPAM Workshop Tutorial

  30. Outline • Part I - Modeling topology • Background • Survey of models + what is known about topology • Example: mathematical foundations • Evaluation of topologies • Part II - Reality check • Beyond simple topology • Visualization • Open questions/Bold statements/Random thoughts • Reading list IPAM Workshop Tutorial

  31. Semantics: policy-based routes • Internet routes are not hop-based shortest paths • General policies: • path between two nodes in a domain remains in that domain • path between two nodes in two different domains traverses zero or more transit domains IPAM Workshop Tutorial

  32. Transit-stub • Use edge weights so that shortest-paths obey general policies • Four weights (in order) • intra-domain edges • T-T edges • S-T edges • S-S edges IPAM Workshop Tutorial

  33. BGP peering relationships (Gao 2000) • Problem: Routes determined by routing policy, including AS-level contractual agreements • Idea: label edges in AS-level graph as • provider-to-customer (customer pays provider for connectivity to rest of Internet) • peer-to-peer (exchange traffic between customers free of charge) • sibling-to-sibling (provide connectivity to rest of Internet for each other) • Use BGP routing table entries AS1 AS7 AS2 AS3 AS6 AS4 AS5 IPAM Workshop Tutorial

  34. Principles • e.g., routing table entry = AS path 1849 702 701 1 • downhill path: all edges provider-to-customer or sibling-to-sibling • uphill path: all edges customer-to-provider or sibling-to-sibling • An AS path of a BGP routing table is: • an uphill path followed by a downhill path (either path segment may be empty)…or... • an uphill path followed by a peer-to-peer edge followed by a downhill path (either path segment may be empty) IPAM Workshop Tutorial

  35. Examples • an uphill path followed by a downhill path • AS4-AS2-AS1-AS3-AS5 • AS7-AS1-AS2 • an uphill path followed by a peer-to-peer edge followed by a downhill path • AS5-AS6-AS3-AS5 • AS6-AS3-AS2-AS4 AS1 AS7 AS2 AS3 AS6 AS4 AS5 IPAM Workshop Tutorial

  36. Basic algorithm sketch • Compute degrees for each AS • For each routing table path: • find highest degree AS (“top provider” T) • AS edge (u,v) to left of T assigned value 1 • AS edge (u,v) to right of T assigned value 1 • For each edge (u,v): • if (u,v) =1 and (v,u) = 1 then sibling-to-sibling • else if (v,u) = 1 then provider-to-customer • else if (u,v) = 1 then customer-to-provider • Note: complete algorithm also identifies peer-to-peer edges IPAM Workshop Tutorial

  37. Hierarchical classification (Subramanian 2002) • Idea: partition ASes into hierarchical levels using directed graph of peering relationships • Process: • identify and remove nodes with out-degree 0 (customers) • recursively identify and remove nodes with out-degree 0 (small ISPs) • identify dense core as largest subset of nodes that is “almost a clique” (in and out-degree at least half nodes) • identify transit core as smallest subset of nodes that peer primarily with each other and ASes in dense core • remaining nodes are outer core IPAM Workshop Tutorial

  38. Example result • Dense core - 20 ASes • Transit core - 162 ASes • Outer core - 675 ASes • Small regional ISPs - 950 ASes • Customers - 8852 ASes IPAM Workshop Tutorial

  39. Outline • Part I - Modeling topology • Background • Survey of models + what is known about topology • Example: mathematical foundations • Evaluation of topologies • Part II - Reality check • Beyond simple topology • Visualization • Open questions/Bold statements/Random thoughts • Reading list IPAM Workshop Tutorial

  40. Visualization: netvisor (Eagan 2002) • Tool for router-level layout • Combines automatic placement with user-assisted placement • Understands domain semantics • Collaboration between Information Visualization experts and Networking experts IPAM Workshop Tutorial

  41. IPAM Workshop Tutorial

  42. Visualization: conceptual model (Faloutsos 2002) • Idea: simple representation of AS-level topology, useful for intuitive understanding (and NY Times publication!) • e.g., bowtie model for web • jellyfish model • highly connected core • layers (“shells”) • degree one nodes form legs • length of legs denotes density layers core legs IPAM Workshop Tutorial

  43. Outline • Part I - Modeling topology • Background • Survey of models + what is known about topology • Example: mathematical foundations • Evaluation of topologies • Part II - Reality check • Beyond simple topology • Visualization • Open questions/Bold statements/Random thoughts • Reading list IPAM Workshop Tutorial

  44. Open Problems • Evaluation • what metrics are important? • Useful modeling/scaling • what topologies should be used for simulations? • Semantics • let’s move beyond simple topology IPAM Workshop Tutorial

  45. Are AS-level topologies useful? • Many interesting problems arise due to large scale of Internet, hence need simulations that are “big enough” • AS-level topology (about 10,000 nodes) manageable for some simulations • But…representation of every AS as a comparable node (especially in 2-d space!) is a gross simplification IPAM Workshop Tutorial

  46. Observations on level of detail • AS level models are limited (useless?) • not enough distinction (all ASes look alike) • not suitable for packet level simulations • router level models are limited (useless?) • too small to be realistic…or... • too large for simulations • need alternative models • intermediate (border routers, exchange points,…) • fluid flow network model?? • need better understanding of scaling IPAM Workshop Tutorial

  47. Reading List (1 of 3) • [Broido 2001] Broido and Claffy, “Internet topology: local properties”, SPIE ITCom 2001. • [Bu 2002] Bu and Towsley, “Distinguishing between Internet power-law generators”, IEEE Infocom 2002. • [Burch 1999] Burch and Cheswick, “Mapping the Internet”, IEEE Computer, April 1999. • [Chen 2002] Chen, Chang, Govindan, Jamin, Shenker and Willinger, “The origin of power laws in Internet topologies revisited”, • [Calvert 1997] Calvert, Doar and Zegura, “Modeling Internet topology”, IEEE Communications Magazine, June 1997. • [Doar 1997] Doar and Leslie, “How bad is naïve multicast routing”, IEEE Infocom 1993. • [Eagan 2002] Netvisor. http://www.cc.gatech.edu/gvu/ii/netviz/ • [Faloutsos 1999] Faloutsos, Faloutsos and Faloutsos, “On power-laws relationships of the Internet topology”, ACM Sigcomm 1999. IPAM Workshop Tutorial

  48. Reading List (2 of 3) • [Gao 2000] Gao, “On inferring autonomous system relationships in the Internet”, IEEE Infocom 2000. • [Govindan 2000] Govindan and Tangmunarunkit, “Heuristics for Internet map discovery”, IEEE Infocom 2000. • [Jin 2000] Jin, Chen and Jamin, “Inet: Internet topology generator”, U. Michigan technical report CSE-TR-433-00, September 2000. • [Medina 2000] Medina, Matta and Byers, “On the origin of power-laws in Internet topologies”, ACM CCR, April 2000. • [Mihail 2002] Mihail, Gkantsidis, Saberi, Zegura, “On semantics of Internet topologies”, GT technical report, January 2002. • [Subramanian 2002] Subramanian, Agarwal, Rexford and Katz, “Characterizing the Internet from multiple vantage points”, IEEE Infocom 2002. • [Tauro 2002] Tauro, Palmer, Siganos and Faloutsos, “A simple conceptual model for the Internet topology”, Global Internet 2001. IPAM Workshop Tutorial

  49. Reading List (3 of 3) • [Tangmunarunkit 2001] Tangmunarunkit, Govindan, Jamin, Shenker and Willinger, “Network topologies, power laws, and hierarchy”, USC technical report 01-746, 2001. • [Waxman 1988] Waxman, “Routing of multipoint connections”, IEEE JSAC, 1988. • [Zegura 1997] Zegura, Calvert and Donahoo, “A quantitative comparison of graph-based models for Internet topology”, IEEE/ACM Transactions on Networking, December 1997. IPAM Workshop Tutorial

  50. The End IPAM Workshop Tutorial

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