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Spreading random connection functions
Massimo Franceschetti
Newton Institute for Mathematical Sciences
April, 7, 2010joint work with
Mathew Penrose and Tom Rosoman
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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
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Bond percolation on the square grid
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The holy grail
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Site percolation on the square grid
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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
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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
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Poisson distribution of points of density λpoints within unit range are connected
S
D
Gilbert graph
A continuum version of a percolation model
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Simplest communication model
A connected component represents nodes which can reach each other along a chain of successive relayed communications
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The critical density
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The critical density
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Random Connection Model
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Simple model for unreliable communication
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Question
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The expected node degree is preserved but connections are spatially stretched
Spreading transformation
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Weak inequality
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Proof sketch of weak inequality
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Strict inequality
It follows that the approach to this limit is strictly monotone from above and spreading is strictly advantageous for connectivity
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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
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Proof sketch of strict inequality
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Spread-out annuli
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Mixture of short and long edges
Edges are made all longer
Spread-out visualisation
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Spread-out dimension
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Open problems
Monotonicity of annuli-spreading and dimension-spreading Monotonicity of spreading in the discrete setting
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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!