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
There is also a brief "data" paper
but I would like someone to continue this style of data collection.
Two less central technical
Somewhat related are
Percolating paths through random points
Optimal flow through the disordered lattice.
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
and the details are in a long technical paper
Analogous to Wikipedia's nice "zooming in"
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