Let me give you some context first: just a few days ago I found some intriguing references to Ricci flows in the setting of directed graphs.
There are at least two versions of Ricci curvature in the discrete realm (one being the Ollivier-Ricci curvature, the other the Forman-Ricci, see here for reference (*)), and as it turns out, they are both useful in graph analytics.
To be a tad more specific, one application leads to a new method for determining communities (the so-called Ricci communities, for the interested ones there is even a github Python implementation which can easily be used for hands-on explorations ), whereas another quite useful one is used to get rid of "bottlenecks" in graph messaging ( thereby solving some critical issue in Graph deep learning see picture below).
https://towardsdatascience.com/over-squashing-bottlenecks-and-graph-ricci-curvature-c238b7169e16
Now, if I understand them correctly, the associated Ricci flow, just like in the differentiable realm, acts as a kind of "curvature heat-like operator", a diffusion which tends to smoothen out the curvature across the underlying geometrical object.
Perhaps naively, it occurred to me this:
why confining ourselves to diffusion? (note: I am aware of the centrality of the Ricci flow in the proof of the Poincare conjecture)
Could one replace the Ricci flow with some kind of PDE (or a difference equation in the finite setting) for the curvature change modeled on completely different PDEs?
For instance, what about a kind of wave equation?
Now the questions (and I apologize if this is too naive, I am coming from the data science world, my knowledge of Riemannian geometry does not go beyond standard grad courses):
- Have such curvature flow involving non-heat-like PDEs been investigated in the world of Riemannian geometry? I would think the answer is in the affirmative, but I just do not happen to know it.
- Are there any references for generalized curvature flows in discrete metric spaces and particularly in weighted directed graphs?
Any help is most welcome.
(*) actually in the referenced article there are three discrete Ricci curvatures, but I haven't wrapped my mind around the third one yet.
Best Answer
A small number of authors have considered hyperbolic versions of the standard flows, see e.g. "Wave character of metrics and hyperbolic geometric flow" by De-Xing Kong and Kefeng Liu and related articles.
I am not familiar with the literature on discrete curvature flows. Some papers that look interesting are "Super Ricci flows for weighted graphs" by Matthias Erbar & Eva Kopfer and "Simplicial Ricci flow" by Warner Miller, Jonathan McDonald, Paul Alsing, David Gu & Shing-Tung Yau. I would suspect that undirected graphs are more natural here than directed graphs, just based on analogy to Riemannian metrics.