Spreading random connection functions

Massimo Franceschetti

Newton Institute for Mathematical Sciences

April, 7, 2010joint work with

Mathew Penrose and Tom Rosoman

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

Bond percolation on the square grid

The holy grail

Site percolation on the square grid

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

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

Poisson distribution of points of density λpoints within unit range are connected

S

D

Gilbert graph

A continuum version of a percolation model

Simplest communication model

A connected component represents nodes which can reach each other along a chain of successive relayed communications

The critical density

The critical density

Random Connection Model

Simple model for unreliable communication

Question

The expected node degree is preserved but connections are spatially stretched

Spreading transformation

Weak inequality

Proof sketch of weak inequality

Strict inequality

It follows that the approach to this limit is strictly monotone from above and spreading is strictly advantageous for connectivity

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

Proof sketch of strict inequality

Spread-out annuli

Mixture of short and long edges

Edges are made all longer

Spread-out visualisation

Spread-out dimension

Open problems

Monotonicity of annuli-spreading and dimension-spreading Monotonicity of spreading in the discrete setting

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!