Wandering around a fibrous network when all the paths look very much alike
I will give an overview of work I have been interested in for the last decade or so, concerned with generating models based on Poisson processes which supply insight into route-finding and traffic flows. Initial constructions (Aldous 2014, Kahn 2016, Kendall 2017) showed existence of such models satisfying the SIRSN axioms formulated by Aldous (2014). But these constructions all involve infinitely long linear paths, and in particular do not easily admit local influences. Can one do better? I shall discuss recent work showing that SIRSN can indeed arise under far less stringent conditions, based on line segments or even suitably stiff fibres.
Aldous, D J. "Scale-Invariant Random Spatial Networks." Electronic Journal of Probability 19, no. 15 (April 2014): 1-41. https://doi.org/10.1214/EJP.v19-2920.
Kahn, J. "Improper Poisson Line Process as SIRSN in Any Dimension." Annals of Probability 44, no. 4 (July 2016): 2694-2725. https://doi.org/10.1214/15-AOP1032.
WSK "From Random Lines to Metric Spaces." Annals of Probability 45, no. 1 (2017): 469-517. https://doi.org/10.1214/14-AOP935.