1. ABIMarch 1. 2007, Espoo1 On the robustness of power law random graphs Hannu Reittu in collaboration with Ilkka Norros, Technical Research Centre of Finland (Valtion Teknillinen Tutkimuskeskus, VTT)

2. ABIMarch 1. 2007, Espoo2 Content  Model definition  Asymptotic architecture  The core  Robustness of the core  Main result and a sketch of proof  Corollaries  Conjecture  Resume

3. ABIMarch 1. 2007, Espoo3 References Norros & Reittu, Advances in Applied Prob. 38, pp.59-75, March 2006 Related models and review: Janson-Bollobás-Riordan, http://www.arxiv.org/PS_cache/math/pdf/0 504/0504589.pdf R Hofstad: http://www.win.tue.nl/~rhofstad/NotesRGC N.pdf

4. ABIMarch 1. 2007, Espoo4 Classical random graph ( )  Independent edges with equal probability (p N ) pNpN pNpN 1-p N

5. ABIMarch 1. 2007, Espoo5 However,  => degrees ~ Bin(N-1, p N ) ≈ Poisson(Np N )  Internets autonomous systems graph (and many others) have power law degrees  Pr(d>k) ~ k -  With 2 < < 3

6. ABIMarch 1. 2007, Espoo6

7. ABIMarch 1. 2007, Espoo7 Conditionally Poissonian random graph model Sequence of i.i.d., >0,r.v. (the ‘capacities’) number of edges between nodes i and j:

8. ABIMarch 1. 2007, Espoo8 Properties, conditionally on : (i) (ii) (iii) The number of edges between disjoint pairs of nodes are independent

9. ABIMarch 1. 2007, Espoo9  Assume :

10. ABIMarch 1. 2007, Espoo10 Theorem (Chung&Lu; Norros&Reittu):  a.a.s. has a giant component  distance in giant component has the upper bound:, almost surely for large N

11. ABIMarch 1. 2007, Espoo11 Asymptotic architecture  Hierarchical layers:

12. ABIMarch 1. 2007, Espoo12 The ‘core’:

13. ABIMarch 1. 2007, Espoo13 ‘Tiers’: Short (loglog N) paths: Routing in the core: next step to largest degree neighbour

14. ABIMarch 1. 2007, Espoo14 The core  ‘Achilles heel’?

15. ABIMarch 1. 2007, Espoo15 Typical path in the ‘core’ WjWj W j-1 W j-2 i*i*

16. ABIMarch 1. 2007, Espoo16 U j-1 is destroyed WjWj W j-1 W j-2 i*i* X X X

17. ABIMarch 1. 2007, Espoo17 Hypothesis:  has a sub graph, a classical random graph with constant diameter,

18. ABIMarch 1. 2007, Espoo18 Back up WjWj W j-1 W j-2 i*i* X X X

19. ABIMarch 1. 2007, Espoo19 hop counts:  a.a.s. Wj

20. ABIMarch 1. 2007, Espoo20  d j is a constant => asymptotically, the same distance ( )

21. ABIMarch 1. 2007, Espoo21 Proposition:  Fix integer j>0  a.a.s., diam(W j )

22. ABIMarch 1. 2007, Espoo22 Remarks  Back up path in W j has at most d j hops  However, in classical random graph, short paths are hard to find  W j is connected sub graph ('peering')

23. ABIMarch 1. 2007, Espoo23 Sketch of proof:  Use the following result (see: Bollobás, Random Graphs, 2nd Ed. p 263, 10.12)  Suppose that functions and satisfy and Then a.e. (cl. random graph) has diameter d

24. ABIMarch 1. 2007, Espoo24  We have:

25. ABIMarch 1. 2007, Espoo25  Find such d: and => the claim follows

26. ABIMarch 1. 2007, Espoo26 Corollaries  Nodes with are removed => extra steps (u.b.). More precisely:

27. ABIMarch 1. 2007, Espoo27 Can we proceed:

28. ABIMarch 1. 2007, Espoo28 Yes and no  If goes to 0 no quicker that:  With this speed

29. ABIMarch 1. 2007, Espoo29 but  Is too quick!  These tiers are not connected because degrees are too low.

30. ABIMarch 1. 2007, Espoo30 Conjecture  However, has a giant component  And degrees =>  Diameter of g.c. (Chung and Lu 2000), yields u.b.

31. ABIMarch 1. 2007, Espoo31 Resume  Removal of ‘large nodes’ has, eventually, no effect on asymptotic distance up to some point  We can imagine graceful growth in path lengths:  Core ( ) is important! Although:  in cl. random graphs, such events do not matter

32. ABIMarch 1. 2007, Espoo32 Thank You!