Internet Topology


17 October 2008
        CS5229, Semester 1, 2008/09
                                                  ...
17 October 2008
   CS5229, Semester 1, 2008/09
                                             
   2
                        ...
Can we characterize the
      Internet’s Topology?


17 October 2008
   CS5229, Semester 1, 2008/09
                      ...
How to generate
        realistic Internet
    topology for simulations?

17 October 2008
   CS5229, Semester 1, 2008/09
 ...
Model Internet quot;
                    as a Graph


17 October 2008
      CS5229, Semester 1, 2008/09
                  ...
Router-Level,quot;
                    node = routerquot;
                   edge = 1-hop link

17 October 2008
       CS5...
AS-Level,quot;
               node = AS domainquot;
                edge = Peering

17 October 2008
    CS5229, Semester 1...
Generating Random
                  Graph


17 October 2008
   CS5229, Semester 1, 2008/09
                               ...
Randomly generate points on a plane




17 October 2008
   CS5229, Semester 1, 2008/09
                                   ...
Connects two nodes with fixed
                   probability p



                                                         ...
Waxman’s Method


17 October 2008
   CS5229, Semester 1, 2008/09
                                             
   11
     ...
Randomly generate points on a plane




17 October 2008
   CS5229, Semester 1, 2008/09
                                   ...
Connects two points with 
                   probability P(u,v)



                                                       ...
Model locality but not
     the structure of Internet


17 October 2008
   CS5229, Semester 1, 2008/09
                   ...
Transit-Stub Method


17 October 2008
   CS5229, Semester 1, 2008/09
                                             
   15
 ...
Randomly generate a graph using 
            Waxman’s method




17 October 2008
      CS5229, Semester 1, 2008/09
       ...
Each node is expanded to form a 
            random graph (transit domain)




17 October 2008
      CS5229, Semester 1, 2...
Connect stub domains to the transit
            domain.




17 October 2008
      CS5229, Semester 1, 2008/09
            ...
Looks good, but is it
        close to the real thing?


17 October 2008
   CS5229, Semester 1, 2008/09
                  ...
“On Power-Law
        Relationships of the
        Internet Topology”quot;
      The Faloutsos brothers,
           SIGCOM...
Use four traces of
         Internet topology
     collected between 97-98

17 October 2008
   CS5229, Semester 1, 2008/09...
AS-Level Topology
 Time
   Num of     Num of       Max       Average
         Nodes
      Edges
    outdegree
 outdegree
N...
Observations: the graphs
    can be decomposed into
     two components: trees
           and core.

17 October 2008
   CS...
Connect stub domains to the transit
            domain.




17 October 2008
      CS5229, Semester 1, 2008/09
            ...
40-50%!
     of the nodes are in trees

17 October 2008
    CS5229, Semester 1, 2008/09
                                  ...
3!
     maximum depth of trees

17 October 2008
   CS5229, Semester 1, 2008/09
                                           ...
1!
           depth of >80% of the
                  trees

17 October 2008
   CS5229, Semester 1, 2008/09
               ...
Time
          Num of     Num of              Max       Average
                   Nodes
      Edges
           outdegree
...
Let quot;
    dv be the out-degree of a
    node, and quot;
    rv be the rank of a node
    (i.e., index in the order of
...
Plot dv versus rv on quot;
                  log-log scale


17 October 2008
    CS5229, Semester 1, 2008/09
             ...
31
32
33
34
Rank Exponent




                 35
Lemma 1:




            36
Lemma 2:




            37
Let quot;
    fd be the number of
    nodes with out-degree dquot;


17 October 2008
   CS5229, Semester 1, 2008/09
      ...
Plot fd versus d on quot;
                      log-log scale


17 October 2008
         CS5229, Semester 1, 2008/09
     ...
40
41
42
43
44
“few connects to many,quot;
           many connects to few”


17 October 2008
   CS5229, Semester 1, 2008/09
            ...
46
Let quot;
    P(h) be the number of
    node pairs within h hops
    of each otherquot;
    (include self-pairs, count
   ...
P(0) = N



17 October 2008
   CS5229, Semester 1, 2008/09
                                             
   48
           ...
P(1) = N + 2E



17 October 2008
     CS5229, Semester 1, 2008/09
                                               
   49
  ...
P(δ) =              N2quot;

    where δ is the diameter of the graph




17 October 2008
   CS5229, Semester 1, 2008/09
 ...
Plot P(h) versus h on quot;
                log-log scale


17 October 2008
   CS5229, Semester 1, 2008/09
               ...
52
53
54
55
56
Average Number of Nodes within h Hops




                                         57
                                    ...
Average Number of Nodes within h Hops 
(using average degree)




                                          58
           ...
59
60
It holds in 97-98.quot;
                   What about later?


17 October 2008
       CS5229, Semester 1, 2008/09
        ...
“Power-Laws and the AS-
      Level Internet Topology”quot;
    G. Siganos and the Faloutsos
     brothers, IEEE/ACM TON

...
63
64
65
Do topologies generated by
     Waxman and Transit-Stub
        exhibit Power Law?


17 October 2008
   CS5229, Semester 1...
67
How to generate topology
        that follows power laws?


17 October 2008
   CS5229, Semester 1, 2008/09
               ...
Where does power law
               comes from?


17 October 2008
   CS5229, Semester 1, 2008/09
                         ...
70
Scientific American, “Scale-Free Networks”, May 2003




                                                       71
        ...
72
73
Generating Power Law
             Topology (simplified)


17 October 2008
   CS5229, Semester 1, 2008/09
                  ...
“On the Original of Power
    Laws in Internet Topologies”quot;
     A Medina, I Matta, J Byers,quot;
       ACM SIGCOMM, ...
76
Randomly generate a small graph




17 October 2008
      CS5229, Semester 1, 2008/09
                                    ...
Incremental Growth: 
                    Add one node at a time




17 October 2008
         CS5229, Semester 1, 2008/09
 ...
Preferential Attachment:
  Connects to a neighbor i with a probability



                                                ...
80
81
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CS5229 Lecture 9: Internet Topology

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CS5229 Lecture 9: Internet Topology

  1. 1. Internet Topology 17 October 2008 CS5229, Semester 1, 2008/09 1
  2. 2. 17 October 2008 CS5229, Semester 1, 2008/09 2
  3. 3. Can we characterize the Internet’s Topology? 17 October 2008 CS5229, Semester 1, 2008/09 3
  4. 4. How to generate realistic Internet topology for simulations? 17 October 2008 CS5229, Semester 1, 2008/09 4
  5. 5. Model Internet quot; as a Graph 17 October 2008 CS5229, Semester 1, 2008/09 5
  6. 6. Router-Level,quot; node = routerquot; edge = 1-hop link 17 October 2008 CS5229, Semester 1, 2008/09 6
  7. 7. AS-Level,quot; node = AS domainquot; edge = Peering 17 October 2008 CS5229, Semester 1, 2008/09 7
  8. 8. Generating Random Graph 17 October 2008 CS5229, Semester 1, 2008/09 8
  9. 9. Randomly generate points on a plane 17 October 2008 CS5229, Semester 1, 2008/09 9
  10. 10. Connects two nodes with fixed probability p ? p 17 October 2008 CS5229, Semester 1, 2008/09 10
  11. 11. Waxman’s Method 17 October 2008 CS5229, Semester 1, 2008/09 11
  12. 12. Randomly generate points on a plane 17 October 2008 CS5229, Semester 1, 2008/09 12
  13. 13. Connects two points with probability P(u,v) ? L: maximum distance d(u,v): distance between u and v 17 October 2008 CS5229, Semester 1, 2008/09 13
  14. 14. Model locality but not the structure of Internet 17 October 2008 CS5229, Semester 1, 2008/09 14
  15. 15. Transit-Stub Method 17 October 2008 CS5229, Semester 1, 2008/09 15
  16. 16. Randomly generate a graph using Waxman’s method 17 October 2008 CS5229, Semester 1, 2008/09 16
  17. 17. Each node is expanded to form a random graph (transit domain) 17 October 2008 CS5229, Semester 1, 2008/09 17
  18. 18. Connect stub domains to the transit domain. 17 October 2008 CS5229, Semester 1, 2008/09 18
  19. 19. Looks good, but is it close to the real thing? 17 October 2008 CS5229, Semester 1, 2008/09 19
  20. 20. “On Power-Law Relationships of the Internet Topology”quot; The Faloutsos brothers, SIGCOMM ‘99 17 October 2008 CS5229, Semester 1, 2008/09 20
  21. 21. Use four traces of Internet topology collected between 97-98 17 October 2008 CS5229, Semester 1, 2008/09 21
  22. 22. AS-Level Topology Time Num of Num of Max Average Nodes Edges outdegree outdegree Nov 97 3015 5156 590 3.42 Apr 98 3520 6432 745 3.65 Dec 98 4398 8256 979 3.76 Router-Level Topology 1995 3888 5012 2.57 22
  23. 23. Observations: the graphs can be decomposed into two components: trees and core. 17 October 2008 CS5229, Semester 1, 2008/09 23
  24. 24. Connect stub domains to the transit domain. 17 October 2008 CS5229, Semester 1, 2008/09 24
  25. 25. 40-50%! of the nodes are in trees 17 October 2008 CS5229, Semester 1, 2008/09 25
  26. 26. 3! maximum depth of trees 17 October 2008 CS5229, Semester 1, 2008/09 26
  27. 27. 1! depth of >80% of the trees 17 October 2008 CS5229, Semester 1, 2008/09 27
  28. 28. Time Num of Num of Max Average Nodes Edges outdegree outdegree Nov 97 3015 5156 590 3.42 Apr 98 3520 6432 745 3.65 Dec 98 4398 8256 979 3.76 Out-degree is highly skewed! 17 October 2008 CS5229, Semester 1, 2008/09 28
  29. 29. Let quot; dv be the out-degree of a node, and quot; rv be the rank of a node (i.e., index in the order of decreasing outdegree) 17 October 2008 CS5229, Semester 1, 2008/09 29
  30. 30. Plot dv versus rv on quot; log-log scale 17 October 2008 CS5229, Semester 1, 2008/09 30
  31. 31. 31
  32. 32. 32
  33. 33. 33
  34. 34. 34
  35. 35. Rank Exponent 35
  36. 36. Lemma 1: 36
  37. 37. Lemma 2: 37
  38. 38. Let quot; fd be the number of nodes with out-degree dquot; 17 October 2008 CS5229, Semester 1, 2008/09 38
  39. 39. Plot fd versus d on quot; log-log scale 17 October 2008 CS5229, Semester 1, 2008/09 39
  40. 40. 40
  41. 41. 41
  42. 42. 42
  43. 43. 43
  44. 44. 44
  45. 45. “few connects to many,quot; many connects to few” 17 October 2008 CS5229, Semester 1, 2008/09 45
  46. 46. 46
  47. 47. Let quot; P(h) be the number of node pairs within h hops of each otherquot; (include self-pairs, count every pair twice) 17 October 2008 CS5229, Semester 1, 2008/09 47
  48. 48. P(0) = N 17 October 2008 CS5229, Semester 1, 2008/09 48
  49. 49. P(1) = N + 2E 17 October 2008 CS5229, Semester 1, 2008/09 49
  50. 50. P(δ) = N2quot; where δ is the diameter of the graph 17 October 2008 CS5229, Semester 1, 2008/09 50
  51. 51. Plot P(h) versus h on quot; log-log scale 17 October 2008 CS5229, Semester 1, 2008/09 51
  52. 52. 52
  53. 53. 53
  54. 54. 54
  55. 55. 55
  56. 56. 56
  57. 57. Average Number of Nodes within h Hops 57
  58. 58. Average Number of Nodes within h Hops (using average degree) 58
  59. 59. 59
  60. 60. 60
  61. 61. It holds in 97-98.quot; What about later? 17 October 2008 CS5229, Semester 1, 2008/09 61
  62. 62. “Power-Laws and the AS- Level Internet Topology”quot; G. Siganos and the Faloutsos brothers, IEEE/ACM TON 17 October 2008 CS5229, Semester 1, 2008/09 62
  63. 63. 63
  64. 64. 64
  65. 65. 65
  66. 66. Do topologies generated by Waxman and Transit-Stub exhibit Power Law? 17 October 2008 CS5229, Semester 1, 2008/09 66
  67. 67. 67
  68. 68. How to generate topology that follows power laws? 17 October 2008 CS5229, Semester 1, 2008/09 68
  69. 69. Where does power law comes from? 17 October 2008 CS5229, Semester 1, 2008/09 69
  70. 70. 70
  71. 71. Scientific American, “Scale-Free Networks”, May 2003 71
  72. 72. 72
  73. 73. 73
  74. 74. Generating Power Law Topology (simplified) 17 October 2008 CS5229, Semester 1, 2008/09 74
  75. 75. “On the Original of Power Laws in Internet Topologies”quot; A Medina, I Matta, J Byers,quot; ACM SIGCOMM, ‘00 17 October 2008 CS5229, Semester 1, 2008/09 75
  76. 76. 76
  77. 77. Randomly generate a small graph 17 October 2008 CS5229, Semester 1, 2008/09 77
  78. 78. Incremental Growth: Add one node at a time 17 October 2008 CS5229, Semester 1, 2008/09 78
  79. 79. Preferential Attachment: Connects to a neighbor i with a probability ? 17 October 2008 CS5229, Semester 1, 2008/09 79
  80. 80. 80
  81. 81. 81

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