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

2. Zegura - Mar 2002IPAM Workshop Tutorial2 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

3. Zegura - Mar 2002IPAM Workshop Tutorial3 Networking background access networks hosts/endsystems routers domains/autonomous systems exchange point stub domains transit domains border routers peering lowly worm

4. Zegura - Mar 2002IPAM Workshop Tutorial4 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

5. Zegura - Mar 2002IPAM Workshop Tutorial5 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

6. Zegura - Mar 2002IPAM Workshop Tutorial6 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 v u d(u,v)

7. Zegura - Mar 2002IPAM Workshop Tutorial7 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

8. Zegura - Mar 2002IPAM Workshop Tutorial8 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

9. Zegura - Mar 2002IPAM Workshop Tutorial9 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

10. Zegura - Mar 2002IPAM Workshop Tutorial10 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

11. Zegura - Mar 2002IPAM Workshop Tutorial11 Router-level measurement 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) source 0 destination 0 S1 D1

12. Zegura - Mar 2002IPAM Workshop Tutorial12 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

13. Zegura - Mar 2002IPAM Workshop Tutorial13 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)

14. Zegura - Mar 2002IPAM Workshop Tutorial14 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

15. Zegura - Mar 2002IPAM Workshop Tutorial15 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

16. Zegura - Mar 2002IPAM Workshop Tutorial16 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 v arbitrary

17. Zegura - Mar 2002IPAM Workshop Tutorial17 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

18. Zegura - Mar 2002IPAM Workshop Tutorial18 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

19. Zegura - Mar 2002IPAM Workshop Tutorial19 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 v s t u v s t

20. Zegura - Mar 2002IPAM Workshop Tutorial20 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

21. Zegura - Mar 2002IPAM Workshop Tutorial21 Notes about models Variants on evolutionary models Variants on degree-driven models Appeal of evolutionary Relationship to work on “networks” in general

22. Zegura - Mar 2002IPAM Workshop Tutorial22 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

23. Zegura - Mar 2002IPAM Workshop Tutorial23 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!”

24. Zegura - Mar 2002IPAM Workshop Tutorial24 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

25. Zegura - Mar 2002IPAM Workshop Tutorial25 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 Small world topologies (Bu 2002) v k nodes

26. Zegura - Mar 2002IPAM Workshop Tutorial26 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

27. Zegura - Mar 2002IPAM Workshop Tutorial27 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)

28. Zegura - Mar 2002IPAM Workshop Tutorial28 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

29. Zegura - Mar 2002IPAM Workshop Tutorial29 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

30. Zegura - Mar 2002IPAM Workshop Tutorial30 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

31. Zegura - Mar 2002IPAM Workshop Tutorial31 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

32. Zegura - Mar 2002IPAM Workshop Tutorial32 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

33. Zegura - Mar 2002IPAM Workshop Tutorial33 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 AS2AS6AS3 AS1AS7 AS4AS5

34. Zegura - Mar 2002IPAM Workshop Tutorial34 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)

35. Zegura - Mar 2002IPAM Workshop Tutorial35 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 AS2AS6AS3 AS1AS7 AS4AS5

36. Zegura - Mar 2002IPAM Workshop Tutorial36 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

37. Zegura - Mar 2002IPAM Workshop Tutorial37 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

38. Zegura - Mar 2002IPAM Workshop Tutorial38 Example result Dense core - 20 ASes Transit core - 162 ASes Outer core - 675 ASes Small regional ISPs - 950 ASes Customers - 8852 ASes

39. Zegura - Mar 2002IPAM Workshop Tutorial39 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

40. Zegura - Mar 2002IPAM Workshop Tutorial40 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

41. Zegura - Mar 2002IPAM Workshop Tutorial41

42. Zegura - Mar 2002IPAM Workshop Tutorial42 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 core layers legs

43. Zegura - Mar 2002IPAM Workshop Tutorial43 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

44. Zegura - Mar 2002IPAM Workshop Tutorial44 Open Problems Evaluation –what metrics are important? Useful modeling/scaling –what topologies should be used for simulations? Semantics –let’s move beyond simple topology

45. Zegura - Mar 2002IPAM Workshop Tutorial45 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

46. Zegura - Mar 2002IPAM Workshop Tutorial46 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

47. Zegura - Mar 2002IPAM Workshop Tutorial47 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.

48. Zegura - Mar 2002IPAM Workshop Tutorial48 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.

49. Zegura - Mar 2002IPAM Workshop Tutorial49 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.

50. Zegura - Mar 2002IPAM Workshop Tutorial50 The End