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

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

12. Random Connection Model

13. Simple model for unreliable communication

14. Question

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

16. Weak inequality

17. Proof sketch of weak inequality

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 Staceys family (see Mathews 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. Mixture of short and long edges Edges are made all longer Spread-out visualisation

23. Spread-out dimension

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

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