Bakry–Émery Ricci Curvature Bounds for Doubly Warped Products of Weighted Spaces

Abstract

We introduce a notion of doubly warped product of weighted graphs that is consistent with the doubly warped product in the Riemannian setting. We establish various discrete Bakry–Émery Ricci curvature-dimension bounds for such warped products in terms of the curvature of the constituent graphs. This requires deliberate analysis of the quadratic forms involved, prompting the introduction of some crucial notions such as curvature saturation at a vertex. In the spirit of being thorough and to provide a frame of reference, we also introduce the -doubly warped products of smooth measure spaces and establish -Bakry–Émery Ricci curvature (lower) bounds thereof in terms of those of the factors. At the end of these notes, we present examples and demonstrate applications of warped products with some toy models.

Discrete Ollivier-Ricci curvature

“…where the operator L is the initial velocity of the (not necessarily Feller) process corresponding to the given 1-parameter random walk; (2) (Lipschitz continuity 1) O Ric(x, y) is locally Lipschitz in µ z and in the distance d (as variables); (3) (Lipschitz continuity 2) if µ ε z as in the previous item is locally Lipschitz in ω, then O Ric(x, y) is locally Lipschitz continuous in the arguments ω and in the distance d (as a variable); (4) (minimizer) the limit-free formulation admits a minimizer; (5) (locality) the minimizer can be localized to be supported in K xy ;…”

“…These applications -discovered not so long a go -include the use of Ollivier-Ricci and Bakry-Émery Ricci curvature as indicators of robustness and as tools by which to measure the difference of two networks; the applications have thus far been in the fields of social, biological or financial networks and at a growing rate due to successes achieved by using these methods. For more details, we refer to [5] and the reference therein. The other important application of Ollivier-Ricci curvature and flow is in community detection i.e.…”

Bakry–Émery Curvature Sharpness and Curvature Flow in Finite Weighted Graphs. Implementation

“…The curvature was then reintroduced several times in the setting of graphs, see [5][6][7]. For further research about Bakry-Émery curvature on finite graphs, see, e.g., [8][9][10][11][12][13][14][15][16][17][18][19].…”