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Analyzing Internet Topology for Future Prediction

Explore the structure and evolution of Internet topologies to predict future network designs and behaviors. Investigate spectral analysis to uncover hidden features. Compare 80s, 90s, and modern models to enhance internet weather forecasting capabilities.

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Analyzing Internet Topology for Future Prediction

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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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