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On the robustness of power law random graphs

On the robustness of power law random graphs

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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