I would like to know if this mathematical concept already exists (and if so, what is the name of it?):
The idea is of a "topological fluid". What this means is a set of points and a list of which points are "neighbours" to other points. (In analogy with a fluid where molecules know about their neighbours through molecular forces.) [The neighbour list may or may not also have a distance between 0 and 1 to its neighbour].
These points are not embedded in any space, the geometry comes entirely from the list of connections. We define a distance between 2 points as the shortest number of steps from neighbour to neighbour.
What makes this a fluid is that the points can change their list of neighbours to points a minimum "distance" away (given certain predefined rules). So that points can move about in this "fluid".
Basically this is changing the topology of a graph step by step but keeping the number of nodes the same.
I have been trying to come up with some rules which would allow the graph to stay mostly uniform and the particles to move about in it. But most rules end up with massive holes in the graph! I don't know if any rules are known which would approximate a fluid in this manner?
I think it should be possible as knowing the neighbours of every particle in a fluid, it should be possible to reconstruct that fluid to within reasonable errors.
One idea I had was an action function which is at a minimum when the graph is fairly uniform (with no big holes or tangles), and then define the dynamics as a path of least action.