Today I remembered a thought I had a while ago, which is that our game histories resolve into “shapes” or “networks”, in which each game can be thought of as a connection between the two players.
Shapes can be classified as either

Fixed, meaning that their shape and volume is unchanging

Finite, meaning that they can depend on a particular set of key points remaining valid (ie. certain players remaining active). When a finite shape “decays” it becomes fixed.

Infinite, meaning that no particular point (player) is required to remain active for the shape to continue to grow. Infinite shapes can decay to finite shapes.
Suppose our group is defined as bugcat and his opponents. This is a star shape. Let’s call the number of players in the shape N, and the number of links from a player L. For bugcat, L = N  1; for all other players L = 1. Notice that although the shape can grow as long as bugcat is active, as soon as he becomes inactive the shape becomes fixed.
We can consider another shape, in which every player must have played every other player – ie. L = N  1 for all players, and call this a polygon. In a polygon, every player in the shape must remain active for new players to enter. It will also become progressively harder for the shape to grow, since the bar to entry increases.
Those are both finite shapes. Now let’s look at infinite ones. The simplest infinite shape is a “path”, like the path of a Shusaku number. New players can attach to the front end of the path, themselves becoming the new key point.
We can also consider a variant in which the path is doubleended, expanding in two directions, and advance this idea to any number of the initial connections of the key player (in this diagram, Gia).
A more sophisticated infinite shape is the ring, in which every player has an L of 2. It can be further complicated by demanding that all ring players connect to a player in the centre, making a cartwheel.
Compared to the path, the ring demands two connections to join rather than one; however, there will be more active players in the shape available to be joined to.
Some shapes better define different human relationships. The polygon is a good match for a small Go club. The path is good for, well, a Shusaku path. The star represents one player’s game history, and so on…