Every unweighted graph $G=(V,E)$ induces a discrete metric space $(V,d)$, where the distance function $d$ can be defined as
$ d(x,y)=\text{shortest path between }x\text{ and }y $
Let's consider the following cases:
- $G$ is a path: then the induced metric space has an Euclidean-like geometry
- $G$ is a circle: then the induced metric space has properties of a spherical geometry
- $G$ is a tree: then the induced metric space has properties of a hyperbolic geometry
Now, for the case that $G$ is bipartite. Do you know about any work connecting the induced metric spaces of bipartite graphs to some geometry?
In other words: which manifold can be viewn as a continuous analogue of bipartite graphs?
For instance, complex networks could be shown to be connected to hyperbolic geometry.