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Identifying Internet Topology

Identifying Internet Topology. Polly Huang NTU, EE & INM http://cc.ee.ntu.edu.tw/~phuang. Outline. The problem Background Spectral Analysis Conclusion. The Problem. What does the Internet look like? Routers as vertices Cables as edges Internet topologies as graphs For example….

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Identifying Internet Topology

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  1. Identifying Internet Topology Polly Huang NTU, EE & INM http://cc.ee.ntu.edu.tw/~phuang

  2. Outline • The problem • Background • Spectral Analysis • Conclusion

  3. The Problem • What does the Internet look like? • Routers as vertices • Cables as edges • Internet topologies as graphs • For example…

  4. The Internet, Circa 1969

  5. A 1999 Internet ISP Map [data courtesy of Ramesh Govindan and ISI’s SCAN project]

  6. Back To The Problem • What does the Internet look like? • Equivalent of • Can we describe the graphs • String, mesh, tree? • Or something in the middle? • Can we generate similar graphs • To predict the future • To design for the future • Not a new problem, but

  7. Becoming Urgent • Internet is an ecosystem • Deeply involved in our daily lives • Just like the atmosphere • We have now daily weather reports to determine whether • To carry umbrella • To fly as scheduled • To fish as usual • To fill holes at the construction site

  8. Internet Weather • We need in the future continuous Internet weather reports to determine • When and how much to bit on eBay • Whether to exchange stock online today • Whether to do free KTV over Skype at 7pm Friday night • Whether to stay with HiNet ADSL or switch to So-Net

  9. Nature of the Research • Need to analyze • dig into the details of Internet topologies • hopefully to find invariants • Need to model • formulate the understanding • hopefully in a compact way

  10. Background • As said, the problem is not new! • Three generations of network topology analysis and modeling already: • 80s - No clue, not Internet specific • 90s - Common sense • 00s - Some analysis on BGP Tables • To describe: basic idea and example

  11. The No-clue Era • Heuristic • Waxman • Define a plane; e.g., [0,100] X [0,100] • Place points uniformly at random • Connect two points probabilistically • p(u, v) ~ 1 / e d ; d: distance between u, v • The farther apart the two nodes are, the less likely they will be connected

  12. Waxman Example

  13. The Common-sense Era • Hierarchy • GT-ITM

  14. GT-ITM • Transit • Number • Connectivity • Stub • Number • Connectivity • Transit-stub • Connectivity

  15. Break-through • Faloutsos et al (SIGCOMM 99) • Analyze routing tables • Autonomous System level graphs (AS graphs) • A domain is a vertex • Find power-law properties in AS graphs • Power-law by definition • Linear relationship in log-log plot

  16. ~2500 ASs with degree 1 1 AS with degree ~2000 Node degree Frequency Node degree Rank Two Important Power-laws

  17. The Power-law Era • Models of the 80s and 90s • Fail to capture power-law properties • BRITE • Inet • Won’t show examples • Too big to make sense

  18. BRITE • Barabasi’s incremental model • Create a random core • Incrementally add nodes and links • Connect new link to existing nodes probabilistically • Waxman or preferential • Node degrees of these graphs will magically have the power-law properties

  19. Inet • Fit the node degree power-laws specifically • Generate node degrees with power-laws • Link degrees preferentially at random

  20. Are They Better? • Tangmunarunkit et al (Sigcomm 2002) • Structural vs. Degree-based • Classification • Structural: TS (GT-ITM) • Degree-based: Inet, BRITE, and etc. • Current degree-based generators DO work better than TS.

  21. Which Degree-based is better? • Compare AS, Inet, and BRITE graphs • Take the AS graph history • From NLANR • 1 AS graph per 3-month period • 1998, January - 2001, March

  22. Methodology • For each AS graph • Find number of nodes, average degree • Generate an Inet graph with the same number of nodes and average degree • Generate a BRITE graph with the same number of nodes and average degree • Compare with addition metrics • Number of links • Cardinality of matching

  23. Number of Links Date

  24. Matching Cardinality Date

  25. Summary of Background • Forget about the heuristic one • Structural ones • Miss power-law features • Power-law ones • Miss other features • But what features?

  26. No Idea! • Try to look into individual metrics • Doesn’t help much • What do you expect from a number that describes a whole graph • A bit information here, a bit there • Tons of metrics to compare graphs! • Will never end this way!!

  27. Spectral Analysis Danica Vukadinovic Thomas Erlebach ETHZ, TIK Polly Huang NTU, EE INM

  28. Our Rationale • So power-laws on node degree • Good • But not enough • Take a step back • Need to know more • Try the extreme • Full details of the inter-connectivity • Adjacency matrix

  29. Outline • The problem • Background • Spectral Analysis • Conclusion

  30. Research Statement • Objective • Identify missing features • Approach • Analysis on the adjacency matrix • can re-produce the complete graph from it • To begin with, look at its eigenvalues • Condensed info about the matrix

  31. No Structural Difference Eigenvalues are proportionally larger. # of Eigenvalues is proportionally larger.

  32. Normalization • Normalized adjacency matrix • Normalized Laplacian • Eigenvalues always in [0,2] • Normalized eigenvalue index • Eigenvalue index always in [0,1] • Sorted in an increasing order • Normalized Laplacian Spectrum (nls) Looking at a whole spectrum Thus referred to as spectral analysis

  33. Trees Grids Eigen-values [0,2] Eigen-values [0,2] Eigenvalue Index [0,1] Eigenvalue Index [0,1] Tree vs. Grid

  34. AS vs. Inet Graphs

  35. nls as Graph Fingerprint • Unique for an entire class of graphs • Same structure ~ same nls • Distinctive among different classes of graphs • Different structure ~ different nls • Do have exception but rare

  36. Spectral Analysis • Qualitatively useful • nls as fingerprint • Quantitatively? • Width of horizontal bar at value 1

  37. Width of horizontal bar at 1 • Different in quantity for types of graphs • AS, Inet, tree, grid • Wider to narrower • The width • Defined as Multiplicity 1 • Denoted as mG(1) • An extended theorem • mG(1)  |P| + |I| - |Q|

  38. For a Graph G • P: subgraph containing pendant nodes • Q: subgraph containing quasi-pendant nodes • Inner: G - P - Q • I: isolated nodes in Inner • R: Inner - I (R for the rest)

  39. Physical Interpretation • Q: high-connectivity domains, core • R: regional alliances, partial core • I: multi-homed leaf domains, edge • P: single-homed leaf domains, edge • Core vs. edge classification • A bit fuzzy • For the sake of simplicity

  40. Validation by Examples • Q • UUNET, Sprint, Cable & Wireless, AT&T • Backbone ISPs • R • RIPE, SWITCH, Qwest Sweden • National networks • I • DEC, Cisco, HP, Nortel • Big companies • P • (trivial)

  41. Revisit the Theory • mG(1)  |P| + |I| - |Q| • Correlation • Ratio of the edge components -> • Width of horizontal bar at value 1 • Grid, tree, Inet, AS graphs • Increasingly larger mG(1) • Likely proportionally larger edge components

  42. Evolution of Edge Ratio of nodes in P Ratio of nodes in I The edge components are indeed large and growing Strong growth of I component ~ increasing number of multi-homed domains

  43. Evolution of Core Ratio of nodes in Q Ratio of links in Q The core components get more links than nodes.

  44. Core Connectivity

  45. Observed Features • Internet graphs have relatively largeredge components • Although ratio of core components decreases, average degree of connectivityincreases

  46. Q R I P Towards a Hybrid Model • Form Q, R, I, P components • Average degree -> nodes, links • Radio of nodes, links in Q, R, I, P • Connect nodes within Q, R • Preferential to node degree • Inter-connect R, I, P to the Q component • Preferential to degree of nodes in Q

  47. Our Premise • Encompass both statistical and structural properties • No explicit degree fitting • Not quite there yet, but… do see an end

  48. Conclusion • Firm theoretical ground • nls as graph fingerprint • Ratio of graph edge -> multiplicity 1 • Plausible physical interpretation • Validation by actual AS names • Validation by AS graph analysis • Framework for a hybrid model

  49. On-going Work • Towards a hybrid model • Fast computation of the graph theoretical metrics • Implementation of the graph generator • Topology scaler • Shrink the topology to its smaller equivalent • To make it possible to simulate at all • Nature of the Internet graph structure • The mechanism that gives rise to the power-laws

  50. Questions? Polly Huang NTU, EE & INM http://cc.ee.ntu.edu.tw/~phuang

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