## Dynamic models of scale-invariant spatial networks

For general background
see this 2013
paper amd my discrete spatial networks page.
One can construct a scale-invariant network in the plane by throwing down random points
(as a space-time Poisson point process) and using some rule to connect a new point to the existing network:
the only requirement is that the rule must be scale-invariant, meaning that it depends only on relative distances,
not absolute distances.
Can we find a rule for which the network looks (at least somewhat) like a real road network?
Here are some simulations by Yucheng Wang.

Simulation 1 uses the rule for the
Relative neighborhood graph,
applied at each step to the newly arriving point.
The resulting network looks very unrealistic, in part because there are many near-parallel lines.
Seeking to avoid that issue,
Simulation 2 modifies that rule by applying the new edge only
if no other edge on either point forms an angle < 15 degrees to the new edge.
Simulation 3 adds a new modification: if a new point is close to an existing edge
then that edge may be replaced by two edges via the new point.
Simulation 4 adds another modification:
when a new point arrives, consider possible edges in the RNG, in increasing order of length, but only create a new edge (new node A to existing node B)
if it has length less than 0.7 times the length of the existing route from A to B.

You can find more details on Yucheng Wang's GitHub page.