Is there a discipline in mathematics that expounds upon a certain notion in graph theory. I was told that in classical graph theory, you can move nodes around without changing the graph as long as the connections stay the same.
Can graphs move through an ambient space?
Say the graph was defined in terms of two intersecting families of functions, embedded on a manifold and flowed according to a continuously changing parameter? That is to say, there is a set of functions scaled by different parameter values, all flowing at the same rate through an ambient space, while preserving connections between nodes. The nodes would be placed at the intersections.
Could one define a graph in terms of a family of intersecting functions with parameter $t,$ in $\Bbb R^2$ as follows? Here is an example:
$f_{s,t}(x)=x^{st}$ and $f_{s,t}(1-x)=(1-x)^{st},$
for $x,f\in (0,1)$ and $s\subset\Bbb Q.$
So if $|s|=n,$ then there are $2n$ total equations and $n^2$ nodes. If $|s|=100$ then there are $200$ total equations and $100^2$ nodes.
Equate $f_{s,t}(x)=f_{s,t}(1-x)$ and place a mass at each intersection. Let $t$ be mathematical time. As time flowed, each node would evolve and trace out a geodesic path. In the case of these particular functions, it would be a vertical path.