Discrete spatial networks

Here the central theme is studying different kinds of optimality property for networks over n given points in two-dimensional space. An overview is given by the survey article: Connected Spatial Networks over Random Points and a Route-Length Statistic (with Julian Shun).

The "central" technical papers are

Short-length routes in low-cost networks via Poisson line patterns (with Wilf Kendall).
The Stretch-Length Tradeoff in Geometric Networks: Average Case and Worst Case Study (with Tamar Lando).
Which Connected Spatial Networks on Random Points have Linear Route-Lengths?
The Shape Theorem for Route-lengths in Connected Spatial Networks on Random Points.

There is also a brief "data" paper

A Route-Length Efficiency Statistic for Road Networks (with Alan Choi)

but I would like someone to continue this style of data collection.

Two less central technical papers are:

A quite different but loosely related theory is developed in Optimal Transportation Networks. This modifies the classical Monge-Kantorovich transportation problem by seeking to design a network (with given sources and sinks) to carry the flows, and optimize with respect to cost, defined as follows. The total cost-per-unit-length of transport along an edge is $\phi^\alpha$ where $\phi$ is the total flow on the edge and $0 < \alpha < 1$ is a parameter. Intuitively, this concave cost setup encourages the network to have "freeways" carrying heavy traffic. The literature focusses on existence and topological properties of optimal networks, not on statistical properties of the optimal networks.

Continuum spatial networks

Here I am investigating consequences of assuming scale-invariance, which necessitates working in the continuum. A short overview paper is

True scale-invariant random spatial networks (with Karthik Ganesan)

and the details are in a long technical paper

Scale-Invariant Random Spatial Networks.

Analogous to Wikipedia's nice "zooming in" demonstration of Brownian scaling, Yucheng Wang has made this MP4 demonstration of the emergence of scale-invariance when we grow a network in the plane by adding random points and using a scale-invariant rule for linking them to the existing network.

A remarkable construction via the Poisson line process was introduced by Wilfrid Kendall in From Random Lines to Metric Spaces and furher developed by hin in Rayleigh Random Flights on the Poisson line SIRSN