Scale-invariant Random Spatial Networks (SIRSN) from line patterns

Prof Wilfrid Kendall (University of Warwick)

Frank Adams 2,

It is possible to join up all pairs of planar points by using paths based on a variant of a Poisson line pattern which is dense everywhere, but with speed limits on the lines such that lines with speeds exceeding $v$ form a Poisson line process with density depending on $v$. Remarkably, this can be done in such a way that (a) any point can be reached from any other point in finite mean time, with unique fastest connection, (b) the pattern formed by shortest-time connections between a finite set of points is scale-invariant, (c) the pattern of shortest-time connections between points of an independent Poisson point process has finite mean length intensity. In short, it is a SIRSN [3, 4]. I will survey the work which establishes all this, and discuss relevance to evaluation of spatial networks, traffic in spatial networks [1, 2], and qualitative behaviour of shortest-time paths.

1. WSK. Geodesics and flows in a Poissonian city. Ann. Appl. Prob., 21 (2011), no. 3, 801-842.
2. WSK. Return to the Poissonian City. J. Appl. Prob (2014), 15A, 297-309.
3. WSK (2016). From Random Lines to Metric Spaces. Ann. Prob. (to appear), 46pp.
4. Kahn, J. (2016). Improper poisson line process as SIRSN in any dimension. Ann. Prob. (to appear).

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