A writeup of my findings so far using meta-graph visualization analysis.
So lets dive in -- This is a snapshot of the visualization. Noticeably, there are 3 very large nodes -- these are districting plans that have a ton of valid flips (under rook contiguity and equal population). So actually, each flip is a pair of 2 flips that preserves equal population.
The next thing that this image illustrates is that in the 4x4 rook grid, there are usually low degree nodes near high degree nodes (see the tiny touching the large nodes). In addition, there are more "balanced" clusters in the bottom left and top center, where the degrees of the nodes are more equal. These clusters tend to look alike, mostly they are reflections / rotations of each other, with small variations.
Lastly, the big node in the top right seems like a "split" node. I highlight this one later.
This graph is the same as before, but i've double clicked two nodes that are connected by an edge. In this diagram, an edge is a series of 2 flips (we must do 2 flips to maintain equal population). The flips happen in sequence, for example: Starting from the top grid, we flip the bottom left orange precicnt left (it becomes blue). Then, to balance the populations, we flip the top right blue node to orange.
A path from one district plan to another via this sequence of two flips defines "connected," and therefore there is an edge between these two diagrams.
The part you've been waiting for -- This picture shows the configurations of the three large nodes in the diagram. It's not suprising that these nodes are the most "balanced" configurations. In fact, the two large nodes on either side are just reflections of each other. The most interesting thing here is when we explore the split node (bottom left), it is the four square orientation! I would call this orientation the most "flexible" seed plan, in that it can easily devolve into many different shapes in the space.
This node is interesting in so many ways. Here are just a few questions about the space of districing plans that arise from it
This is one of the other two big nodes pulled out so you can see it's connections and the shape of the graph around it. Definitely play with this stuff!