Bakry-Émery curvature sharpness and curvature flow in finite weighted graphs. I. Theory

Abstract

In this sequence of two papers, we introduce a curvature flow on (mixed) weighted graphs which is based on the Bakry-Émery calculus. The flow is described via a time-continuous evolution through the weighting schemes. By adapting this flow to preserve the Markovian property, its limits turn out to be curvature sharp. Our aim is to present the flow in the most general case of not necessarily reversible random walks allowing laziness, including vanishing transition probabilities along some edges ("degenerate" edges). This approach requires to extend all concepts (in particular, the Bakry-Émery curvature related notions) to this general case and it leads to a distinction between the underlying topology (a mixed combinatorial graph) and the weighting scheme (given by transition rates). We present various results about curvature sharp vertices and weighted graphs as well as some fundamental properties of this new curvature flow.This paper is accompanied by a second paper discussing the curvature flow implementation in Python for practical use. In this second paper we present examples and exhibit further properties of the flow.

Discrete curvature on graphs from the effective resistance*

“…In a similar way, if we left-multiply identity (19) by Ω and use the fact that Ωp = 2σ 2 as follows from ( 22) in the next section, we find…”

“…Many results in this section follow from the identity (19) in appendix A which, we recall, is implied by Fiedler's identity [20]. We repeat the identity for ease of presentation:…”

“…Due to its importance in the continuous setting, there have been several proposals to define a Ricci flow in the case of discrete curvatures, with various applications: Jin et al [41] define a Ricci flow for meshes in computer graphics, Ni et al [63] use a Ricci flow inspired evolution of networks for community detection and network alignment and Weber et al considered Ricci flows related to the Forman Ricci curvature [85]. Recently, Bai et al studied a Ricci flow based on the Lin-Yu-Yau curvature [4] and Cushing et al studied a Ricci flow based on Bakry-Émery curvature [19]. Here, we show that there is a natural Ricci flow associated to the resistance curvature with promising features.…”

“…A first family of definitions for the node resistance curvature follows immediately from identity (19). From the diagonal entries (QΩ) ii we retrieve definition (2) for the node resistance curvature as…”

“…Starting again from identity (19) and right-multiplying with the resistance curvature vector p, we find that QΩp = 0 ⇒ Ωp ∈ ker(Q). Since ker(Q) = span(u) for a connected graph [16,21] this means that Ωp is a constant vector and, including the unit-sum property 17 , that Ωp = 2σ 2 u.…”