How quickly can one get around in a network, compared with travelling at the same speed along a straight-line route? If the network is composed of random straight lines, then the answer is "surprisingly quickly" (Aldous & Kendall, 2008; Kendall, 2013). One can also answer questions about distribution of traffic on the network (Kendall, 2013, 2014b), and even devise more complicated networks which generate scale-invariant random metric spaces (Kendall, 2015; Kahn 2015); in this last case the clue is to produce a dense network of lines, each with associated maximum speed, constructed in a scale-invariant way, and to arrange for most lines to have very low speeds.
A general introduction to this sub-topic of transportation networks is given in Kendall (2014a). You can watch a rather low-quality video of me giving a seminar on this at https://www.youtube.com/watch?v=TyQ5KCyg4vc.
Recent work with my PhD student Gameros has considered traffic in the Poissonian city, building on results in Kendall (2011, 2014b). There are intriguing empirical connections to the statistical work in the very controversial report by Beeching (1963).
A very interesting PhD topic to pursue is the following. Consider the random metric space constructed in Kendall (2014). Can one define and investigate families of diffusions on this space, by analogy to the Liouville diffusions constructed for 2D quantum gravity? I gave a related talk at an LMS Durham Symposium in August 2017. There have been recent significant advances in unpublished work by Kendall & Banerjee, but significant interesting questions remain.
- Aldous, D. J., & Kendall, W. S. (2008). Short-length routes in low-cost networks via Poisson line patterns. Advances in Applied Probability, 40(1), 1–21.
- Beeching, R. (1963). The reshaping of British railways. Her Majesty’s Stationery Office.
- Kahn, J. (2016). Improper Poisson line process as SIRSN in any dimension. Annals of Probability, 44.4, 2694-2725.
- Kendall, W. S. (2011). Geodesics and flows in a Poissonian city. Annals of Applied Probability, 21(3), 801–842.
- Kendall, W. S. (2014a). Lines and networks. Markov Processes and Related Fields, 20(1), 81–106.
- Kendall, W. S. (2014b). Return to the Poissonian City. Journal of Applied Probability, 15A, 297-309.
- Kendall, W. S. (2015). From Random Lines to Metric Spaces. Annals of Probability, to appear, 46pp.