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Spreading random connection functions

Spreading random connection functions. Massimo Franceschetti. Newton Institute for Mathematical Sciences April, 7, 2010 joint work with Mathew Penrose and Tom Rosoman. The result in a nutshell.

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Spreading random connection functions

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  1. Spreading random connection functions Massimo Franceschetti Newton Institute for Mathematical Sciences April, 7, 2010 joint work with Mathew Penrose and Tom Rosoman

  2. The result in a nutshell In networks generated by Random Connection Models in Euclidean space, occasional long-range connections can be exploited to achieve connectivity (percolation) at a lower node density value

  3. Bond percolation on the square grid

  4. The holy grail

  5. Site percolation on the square grid

  6. Still very far from the holy grail Grimmett and Stacey (1998) showed that this inequality holds for a wide range of graphs beside the square grid

  7. Proof of by dynamic coupling Canreach anywhere inside a green site percolation cluster via a subset of the open edges in the edge percolation model The same procedure works for any graph, not only the grid

  8. D S Gilbert graph A continuum version of a percolation model Poisson distribution of points of density λ points within unit range are connected

  9. Simplest communication model A connected component represents nodes which can reach each other along a chain of successive relayed communications

  10. The critical density

  11. The critical density

  12. Random Connection Model

  13. Simple model for unreliable communication

  14. Question

  15. Spreading transformation The expected node degree is preserved but connections are spatially stretched

  16. Weak inequality

  17. Proof sketch of weakinequality

  18. Strict inequality • It follows that the approach to this limit is strictly monotone from above and spreading is strictly advantageous for connectivity

  19. Main tools for the proof of The key technique is ‘enhancement’ Menshikov (1987), Aizenman and Grimmett (1991), Grimmett and Stacey (1998) • We also need the inequality for RCM graphs which are not included in Grimmett and Stacey’s family (see Mathew’s talk on Friday) • And use of a dynamic construction of the Poisson point process and some scaling arguments

  20. Proof sketch of strict inequality

  21. Spread-out annuli

  22. Spread-out visualisation Mixture of short and long edges Edges are made all longer

  23. Spread-out dimension

  24. Open problems • Monotonicityof annuli-spreading and dimension-spreading Monotonicity of spreading in the discrete setting

  25. Conclusion In real networks spread-out long-range connections can be exploited to achieve connectivity at a strictly lower density value Main philosophy is to compare different RCM percolation thresholds rather than search for exact values in specific cases Thank you!

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