Bakry-Émery curvature on graphs as an eigenvalue problem

Abstract

In this paper, we reformulate the Bakry-Émery curvature on a weighted graph in terms of the smallest eigenvalue of a rank one perturbation of the so-called curvature matrix using Schur complement. This new viewpoint allows us to show various curvature function properties in a very conceptual way. We show that the curvature, as a function of the dimension parameter, is analytic, strictly monotone increasing and strictly concave until a certain threshold after which the function is constant. Furthermore, we derive the curvature of the Cartesian product using the crucial observation that the curvature matrix of the product is the direct sum of each component. Our approach of the curvature functions of graphs can be employed to establish analogous results for the curvature functions of weighted Riemannian manifolds. Moreover, as an application, we confirm a conjecture (in a general weighted case) of the fact that the curvature does not decrease under certain graph modifications.

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

“…Examples of non-degenerate curvature sharp weighted graphs are simple random walks without laziness on complete graphs K n (with constant curvature K ∞ (x) = 1 2 + 3 2(n−1) ) or simple random walks without laziness on triangle-free d-regular graphs (with K ∞ (x) ≤ 2 d ). Curvature sharpness was originally introduced in [CLP20] for the non-normalized Laplacian on combinatorial graphs and in [CKLP22] for general weighted graphs. The curvature sharpness definitions in these papers differ from the one given here.…”

“…Following ideas in [CKLP22] (see also [Sic21] for the unweighted case), the curvature K ∞ (x) of a non-degenerate vertex x ∈ V can also be expressed as the smallest eigenvalue of a particular symmetric matrix A ∞ (x) = A P,∞ (x) of size d x , the number of outgoing edges from x. The matrix A ∞ (x) is called the curvature matrix at x ∈ V , and we view it as a discrete version of the Ricci curvature tensor acting as a quadratic form on the tangent space T x M in the smooth setting of a Riemannian manifold (M, g).…”

“…This symmetric matrix Q is uniquely determined by the relation v Q(x)v = min Γ 2 (f ), where the minimum runs over all f : V → R with f (x) = 0 and f = v on S 1 (x) (see Proposition 2.14). While A N (x) exists only in the case of a non-degenerate vertex x (since its derivation requires to divide the (i, j)-th entries of 2Q(x) by the transition rate expressions √ p xy i p xy j , see [CKLP22,(1.2)]), the matrix Q(x) can also be defined for degenerate vertices, and the identity in (3) of Theorem 1.3 can then be viewed as a generalisation of the curvature sharpness description (8) for non-degenerate weighted graphs to the case of weighted graphs with both degenerate and non-degenerate vertices. Before we discuss other results, let us briefly reflect on the equivalences in Theorem 1.3: Recall from Definition 1.2 that a vertex is called curvature sharp if it is curvature sharp for some dimension N ∈ (0, ∞], that is, the Bakry-Émery curvature K N (x) agrees with the upper curvature bound…”

“…Curvature reformulation via the Schur complement and the matrix Q(x). In this subsection, we revisit the core concept in [CKLP22], which is to reformulate the Bakry-Émery curvature. With a subtle modification, we extend the reformulation of the curvature (which was defined only for a non-degenerated weighted graph) to be valid for a degenerated weighted graph.…”

“…Fortunately, the equivalence "(26) ⇔ (27)"extends to the case when Γ 2 (x) S 2 is only positive semidefinite; see Proposition 2.11 below. Recall that we have (see [CKLP22,(A.8) and (A.9)])…”