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On the robustness of power law random graphs

On the robustness of power law random graphs. Hannu Reittu in collaboration with Ilkka Norros, Technical Research Centre of Finland (Valtion Teknillinen Tutkimuskeskus, VTT). Content. Model definition Asymptotic architecture The core Robustness of the core

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On the robustness of power law random graphs

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

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

  3. 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/0504/0504589.pdf R Hofstad: http://www.win.tue.nl/~rhofstad/NotesRGCN.pdf March 1. 2007, Espoo

  4. Classical random graph ( ) • Independent edges with equal probability (pN) pN pN 1-pN March 1. 2007, Espoo

  5. However, • => degrees ~ Bin(N-1, pN) ≈Poisson(NpN) • Internets autonomous systems graph (and many others) have power law degrees • Pr(d>k) ~ k- • With 2 << 3 March 1. 2007, Espoo

  6. March 1. 2007, Espoo

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

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

  9. Assume: March 1. 2007, Espoo

  10. 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 March 1. 2007, Espoo

  11. Asymptotic architecture • Hierarchical layers: March 1. 2007, Espoo

  12. The ‘core’: March 1. 2007, Espoo

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

  14. The core • ‘Achilles heel’? March 1. 2007, Espoo

  15. Typical path in the ‘core’ i* Wj-2 Wj-1 Wj March 1. 2007, Espoo

  16. Uj-1 is destroyed X i* X Wj-2 X Wj-1 Wj March 1. 2007, Espoo

  17. Hypothesis: • has a sub graph, a classical random graph with constant diameter, March 1. 2007, Espoo

  18. Back up X i* X Wj-2 X Wj-1 Wj March 1. 2007, Espoo

  19. hop counts: • a.a.s. Wj March 1. 2007, Espoo

  20. dj is a constant => asymptotically, the same distance ( ) March 1. 2007, Espoo

  21. Proposition: • Fix integer j>0 • a.a.s., diam(Wj) March 1. 2007, Espoo

  22. Remarks • Back up path in Wj has at most djhops • However, in classical random graph, short paths are hard to find • Wj is connected sub graph ('peering') March 1. 2007, Espoo

  23. 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 March 1. 2007, Espoo

  24. We have: March 1. 2007, Espoo

  25. Find such d: and => the claim follows March 1. 2007, Espoo

  26. Corollaries • Nodes with are removed => extra steps (u.b.). More precisely: March 1. 2007, Espoo

  27. Can we proceed: March 1. 2007, Espoo

  28. Yes and no • If goes to 0 no quicker that: • With this speed March 1. 2007, Espoo

  29. but • Is too quick! • These tiers are not connected because degrees are too low. March 1. 2007, Espoo

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

  31. 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 March 1. 2007, Espoo

  32. Thank You! March 1. 2007, Espoo

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