Perpetual cutoff method and discrete Ricci curvature bounds with exceptions

One of the main obstacles regarding Bakry–Emery curvature on graphs is that the results require a global uniform lower curvature bounds where no exception sets are allowed. We overcome this obstacle by introducing the perpetual cutoff method. As applications, we prove gradient estimates only requiring curvature bounds on parts of the graph. Moreover, we sharply upper bound the distance to the exception set for graphs having uniformly positive Bakry–Emery curvature everywhere but on the exception set.

An important result for Bakry Emery curvature on graphs is the gradient estimate ΓP t f ≤ e −2Kt P t Γf, for the heat semigroup P t = e t∆ , assuming the curvature dimension condition CD(K, ∞) everywhere, see [GL17; HL17; LL15; LP18].As soon as one vertex of the graph behaves badly and has negative curvature, the gradient estimate potentially gets far off from what would expect.For large girth graphs, a geometric cutoff might be possible, i.e., one can hope that induced subgraphs inherit lower curvature bounds.However this appears to be hopeless for small girth graphs since cutting out edges will possibly reduce connectedness within one-spheres which usually decreases curvature.We overcome the obstacle of cutting off an exception set W by introducing an analytic cutoff method inspired by the gradient estimate proofs in [MW19].The idea is to define a modified semigroup P W t staying constant in space (but not in time) on the exception set W , and behaving like the standard semigroup on the part with curvature bound.Staying constant on the exception set makes the curvature there invisible since it cannot contribute to the gradient.Behaving like the standard semigroup on the part with curvature bound allows to employ the Γ-calculus yielding gradient estimates only requiring curvature bounds on parts of the graph, see Section 5.In particular in Theorem 5.1, we prove when assuming CD(K, ∞) only on V \ W .As application, we give distance bounds under non-constant Bakry-Emery curvature, see Section 6.In particular, we obtain sharp distance bounds recovering the Bonnet-Myers diameter bounds from [LMP18].
In Theorem 6.1, we will prove Here, D denotes the maximal vertex degree.This greatly improves a result of [Liu+19] where it was shown where an overall curvature bound Ric ≥ −K 0 is needed, i.e., the distance bound can get arbitrarily far off if the curvature on W behaves badly.In contrast, our result does not depend on the the curvature of W at all and gives the sharp distance bound one would expect from the Bonnet Myers theorem on graphs (see [LMP18, Corollary 2.2]).Finally in Section 7, we give a general discussion on the significance of the new cutoff method and point out further research directions for possible interest.

Setup and notation
We say ) is symmetric and zero on the diagonal, and if m : V → (0, ∞).The function w is called edge weight and m is called vertex measure.We write x ∼ y for x, y ∈ V if w(x, y) > 0. We will only consider locally finite graphs, i.e., #{y ∈ V : y ∼ x} < ∞ for all x ∈ V .The (weighted) vertex degree is given by Deg(x) := y w(x, y)/m(x) for x ∈ V .We will assume that Deg is bounded as a standing assumption in the paper.The combinatorial graph distance d on V is given by d(x, y) := inf{n : x = x 0 ∼ . . .∼ x n = y for some x k ∈ V }.We will always assume that G is connected, i.e., d(x, y) < ∞ for all x, y ∈ V .We define the function spaces Denote by P t = e t∆ the associated heat semigroup.We now introduce the Bakry-Emery calculus following [BGL14; LY10; Sch99] to define Ricci curvature on graphs.
The gradient form Γ : For convenience, we write Γf := Γ(f, f ) and We say G satisfies the curvature dimension condition CD(K, n) This is analog to Bochner's inequality on manifolds and defines the Bakry-Emery curvature on graphs where K ∈ R is a lower curvature bound and n > 0 is an upper dimension bound.We say G satisfies CD(K, n) if it is satisfied at all x ∈ V .The CD(K, n) inequality was initially used as a curvature bound on diffusion semigroups and manifolds in [Bak87; BE85], and later transfered to discrete settings in [LY10;Sch99].For the sake of clarity, we restrict ourselves to the case n = ∞ where the maximum vertex degree in some sense substitutes the notion of dimension.Note that in case of bounded vertex degree Deg ≤ D, one can always trade off the dimension term with a curvature term.In particular, since (∆f ) 2 ≤ 2DΓf due to Cauchy-Schwarz.

Dini Derivatives, Gamma calculus, and semigroups
For dealing with the non-smoothening effects of the cutoff, we define the upper and lower left and right Dini derivatives for functions F : I → R for an interval I via If the limits exist, we write ∂ ± t for the left and right derivative.We next give basic properties of the Dini derivative from [HT06].
As a preparation for the cutoff method, we show compatibility of the right derivatives, the semigroup, and the Γ calculus.
Lemma 3.2.Let G = (V, w, m) be a graph with bounded vertex degree.Let (u t ) t≥0 ∈ ℓ ∞ (V ) be continuous in t w.r.t • ∞ .Suppose the right derivative ∂ + t u t (x) exists everywhere.Then for t > s ≥ 0, follows from the interchangeability of ∂ + t and ∆, and from the product rule Due to continuity of P t , the latter part converges to P t−s ∂ + s u s (x).Due to the mean value theorem, there exists δ = δ(ε) ∈ [0, ε] s.t. the former part equals −∆P t−s−δ u s+ε (x) which, due to continuity of u t and P t , converges to −∆P t−s u s (x) as ε and thus δ(ε) tend to zero.Putting together, and applying commutativity of ∆ and P t on ℓ ∞ (V ) following from bounded vertex degree, gives ∂ + s (P t−s u s ) = P t−s (∂ + s − ∆)u s as desired.

Perpetual cutoff method
The perpetual cutoff method we introduce in this section is inspired by the cutoff method given in [MW19] to prove gradient estimates for Ollivier curvature.The main difference is that in [MW19], the cutoff threshold depends on space, but not on time, whereas in this article, the cutoff threshold only depends on time, but not on space.Let ∅ = W ⊂ V .We define the closure and we define for all w ∈ cl(W )}.This is the set of all functions being constant on cl(W ) and not smaller outside.We define the cutoff operator f.
The reason why we define S W via the closure cl(W ) is that we want the gradient Γf to be zero on W for f ∈ ℓ W ∞ (V ).This turns out to be essential for showing that the gradient estimate is independent of the curvature on W .We next define the single time application of the cutoff We finally come to the most relevant definition of the paper.The (non-linear) perpetual cutoff semigroup is given by where n is arbitrary all t i are positive.We are now prepared to give a collection of useful properties of P W t and its domain ℓ W ∞ (V ).
Theorem 4.1.Let G = (V, w, m) be a graph with bounded vertex degree Deg ≤ D.
(viii) P W ∞ f := lim t→∞ P W t f is constant on V .
We now prove (iv).
Due to the mean value theorem and since Since inf V Q W t f = sup cl(W ) P t f and by the mean value theorem, there exists s ∈ [0, t] s.t.
Putting together with (1) and applying : otherwise and more precisely, for small enough t > 0, Taking supremum yields . This proves (iv).Claim (v) follows immediately from (i) and (iv).We next prove (vi).Due to the semigroup property, it suffices to prove is a contraction.Due to the mean value theorem, there exists 0 ≤ s ≤ t s.t.
) due to continuity of Γ, ∆ and P t .Due to (iv), we have ) which proves (vi).We next prove (vii).Since f is constant on cl(W ), we have Γf = 0 on W and Γ 2 f ≥ 0 on W due to [HL19, Proposition 1.1].
It is left to prove (viii).Obviously, inf V P W t f is non-decreasing, and sup V P W t f is non-increasing in t.Therefore both limits exist when t → ∞.Let ε > 0 and let c := lim t→∞ inf V P W t f and C := lim t→∞ sup V P W t f .We aim to show By applying iteratively and taking supremum, this estimate also holds for P W t f .Applying to g = P W T f yields In particular, P W t g − P t g ∞ ≤ ε.Since P t g converges to a constant as t → ∞, this gives C − c ≤ 2ε.Taking ε → 0 proves claim (viii).This finishes the proof of the theorem.

Gradient estimates
Having settled the machinery of the perpetual cutoff method, we can give a short proof of the gradient estimate requiring a curvature bound not everywhere.
Theorem 5.1.Let G be a graph with bounded weighted vertex degree.Suppose G satisfies CD(K, ∞) on V \ W for some

Distance bounds
Using the gradient estimates, we prove the distance bound analogically to the diameter bounds in [LMP18].Theorem 6.1.Let G be a graph with bounded vertex degree Deg ≤ D. Suppose G satisfies CD(K, ∞) on V \ W for some K > 0 and some ∅ = W ⊂ V .Then, Due to Theorem 4.1 (v), and due to |∆g| 2 ≤ 2DΓg for all g ∈ C(V ), and due to Theorem 5.1, we have where the last estimate follows from Γg ≤ D/2 for every 1-Lipschitz function g ∈ C(V ).Hence for every x ∈ V and w ∈ cl(W ), where we applied Theorem 4.1 (viii) for the identity, and Proposition 3.1 (ii) for the first estimate where the continuity of P W t follows from Theorem 4.1 (iii).This immediately implies the claim of the theorem.

Discussion and outlook
The perpetual cutoff method seems to exhibit a vast versatility.It subtly intertwines with both Ollivier and Bakry-Emery curvature giving interesting new gradient estimates.There is good hope that it also works well both for exponential Bakry-Emery curvature and entropic curvature on graphs.It is conceivable that the perpetual cutoff method gives new insights on non-discrete settings like metric measure spaces, Markov processes or Dirichlet forms.It seems natural to ask whether the gradient estimates and distance bounds can be extended to curvature bounds with finite dimension term.We now point out a different direction for applying the perpetual cutoff method on graphs.Given a graph G = (V, w, m) and an ordered sequence V ⊃ W 1 ⊃ W 2 ⊃ . . .s.t V \ W k is finite and W k = ∅.Under which conditions do we have lim k→∞ P W k t f = P t f ?We remark that the pointwise limit exists since P W k t f is non-increasing in k.A positive answer to this question would make the perpetual cutoff method compatible with the maximum principle which expectedly relies on finiteness of V \ W .
Then, ΓP W t f ≤ e −2Kt P t Γf.Proof.Let F (s) := e 2Ks P t−s (ΓP W s f ).Due to Lemma 3.2 (ii), we can use product and chain rule for calculating the right derivative of F .Thus, − 2e 2Ks P t−s Γ 2 P W s f where the inequality follows from Theorem 4.1 (vi).Observe Γ 2 P W t−s f ≥ KΓP W t−s f on V \W due to CD(K, ∞), and Γ 2 P W t−s f ≥ 0 = KΓP W t−s f on W due to Theorem 4.1 (vii).Putting together yields ∂ + s F (s) ≤ 2KF (s) − 2Ke 2Ks P t−s ΓP W s f = 0. Hence by Proposition 3.1 (i), we see that F (s) is non-increasing which immediately implies the claim of the theorem.