Distance Bounds for Graphs with Some Negative Bakry-Émery Curvature

Abstract We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits an explicit upper bound for the diameter. Otherwise, the graph is a subset of the tubular neighborhood with an explicit radius around the non-positively curved vertices. Those results seem to be the first assuming non-constant Bakry-Émery curvature assumptions on graphs.


Introduction
In Riemannian geometry, diameter bounds for complete connected Riemannian manifolds are well established under several curvature assumptions. The well known Bonnet-Myers theorem states that if the Ricci curvature of a manifold is larger than a positive threshold, the diameter of the manifold is nite, and, therefore, the manifold itself compact, see [24,35] and the references therein. In particular, the classical Jacobi eld technique used there provides also a sharp upper estimate for the diameter. Later on, this result was generalized in [36]. There, the authors assumed that the amount of the Ricci curvature of the manifold M below a positive level is locally uniformly L p -small for some p > dim M/ , and obtain indeed a diameter bound depending on this kind of smallness of the curvature. The concept of Ricci curvature was transferred into various settings. Let us provide a brief summary of the history. Already in 1985, Bakry and Émery introduced Ricci curvature on di usion semigroups via the highly generalizable Γ-calculus [2] derived from the Bochner formula. This approach has rst been applied to a discrete setting in [8] and diversely used in [4,7,15,17,20,21,28,30,32,33,37]. The theory of local metric measure spaces has also bene tted from the Bakry-Émery approach. For more information about Ricci-curvature on metric measure spaces, see [1,10,29,38]. A concept of Ricci curvature on graphs via optimal transport has been introduced by Ollivier [34] and applied in [5,23,27,30]. Recently, Erbar, Maas and Mielke introduced a Ricci curvature on graphs via convexity of the entropy [9,11,12,31]. In a highly celebrated paper, Erbar, Kuwada and Sturm proved that on metric measure spaces, the concepts of Ricci curvature via Γ-calculus (Bakry-Émery) and optimal transport and entropy (Lott-Sturm-Villani) coincide [10]. On the other hand, in the setting of graphs, Bakry-Émery Ricci curvature and Ollivier Ricci curvature are often quite di erent and there are many open questions about the relations between these curvature notions.
It is now natural to ask for analogues and generalizations of the diameter bounds for manifolds above to contexts in which concepts of Ricci curvature exist. For metric measure spaces, there have been attempts to generalize the Bonnet-Myers theorem to variable Ricci curvature bounds in an integral sense, see [25] and the references therein. For connected graphs G = (V , E), the authors of [28] show a sharp diameter bound assuming positive Bakry-Émery curvature in the CD(K, N)-setting for N ∈ ( , ∞], notions of curvature we will introduce below. For convenience, we recall the result for further reference. Theorem 1.1 ([28]). Let G = (V , w, m) be a graph.
1. Assume that CD(K, ∞) holds for K > and the graph admits an upper bound Deg max for the weighted vertex degree. Then, we have where diam d is the diameter of G with respect to the combinatorial distance. 2. Assume that CD(K, N) holds for K, N > . Suppose that G is complete in the sense of [20] and satis es inf x∈V m(x) > . Then, we have where σ is the resistance metric de ned below.
In this article, we generalize the above discrete Bonnet-Myers theorem to the situation where the graph is positively curved except on a vertex set V , where the curvature is allowed to be non-positive. The main result below states that a graph is always covered by the tubular neighborhood around the negatively curved vertices of an explicit radius depending on local curvature dimension assumptions, which are given pointwise by the Bochner formula shown below. This description of the curvature involves the Laplacian of the space considered. The idea is to compare the di erent curvature values on the sets V and V \ V via the semigroups associated to di erent Laplacians. On one hand, we have a graph of constant positive curvature, the lower curvature bound of V \V , and a graph of constant negative curvature, the lower curvature bound of V . Those lead to di erent Laplacians and therefore to di erent semigroups, which have to be controlled in a manner such that the diameter of the whole graph can be bounded above. After we introduced the neccessary framework and the main result in the section below, we show several preparatory estimates of the semigroup depending on the set of negatively curved vertices and re ne the analysis of the techniques developed in [28].

Setting and main result
Let G = (V , w, m) be a weighted, connected, locally nite graph. That is, on the vertex set V ≠ ∅, we introduce a symmetric map If x, y are two vertices with wxy > , we say they are neighbors, or they are connected by an edge, and write x ∼ y. We say G is locally nite if each vertex has nitely many neighbors. The maps w and m introduced above represent the edge measure and the vertex measure of G, respectively. For any two vertices of a connected graph, there is a path connecting them. The graph distance d : V ×V → [ , ∞] is given by the number of edges in a shortest path between two vertices. The diameter diam(V ) of a set V ⊂ V is the maximum graph distance between any two vertices in V . By Tr(V ) we denote the tubular neighborhood of V ⊂ V of radius r > . If V = {x} for some x ∈ V, then Br(x) := Tr(V ), the ball around x with radius r > . As usual, the weighted degree of a vertex x ∈ V is given by We say that G has bounded vertex degree if there exists Deg max < ∞ with Deg(x) ≤ Deg max for all vertices x ∈ V. Denote by Cc(V) the set of nitely supported functions on V and · ∞ the maximum norm. The Laplacian on functions f ∈ Cc(V) is de ned by Remark 2.1. If m(x) = y∈V wxy for any x ∈ V, the associated Laplacian is called the normalized Laplacian. If m(x) = for any x ∈ V, the Laplacian is called combinatorial or physical.
The de nition of the Laplacian leads to the so-called carré du champ operator Γ: for all f , g ∈ Cc(V), x ∈ V: For simplicity, we always write Γ(f ) := Γ(f , f ). Iterating Γ, we can de ne another form Γ , which is given by We abbreviate Γ (f ) = Γ (f , f ). As mentioned before, the graph distance is de ned by shortest paths between two points. In contrast, we can de ne another kind of metric coming from the operator Γ.
For an intrinsic metric ρ, the jump size of ρ is given by (iii) The resistance metric σ on V is given by As in the case of the graph distance, if r is an intrinsic or the resistance metric, we de ne diamr(V ) for a subset V ⊂ V to be the diameter of V with respect to r, and T r R (V ) denotes the tubular neighborhood of V of radius R with respect to r, etc. Intrinsic metrics have already been used to solve various problems on graphs, see, e.g., [3,6,13,14,16,18,19].
Remark 2.4. (i) All metrics smaller than an intrinsic metric are intrinsic, too. In general, the resistance metric σ is not intrinsic, but is greater than all intrinsic metrics.
(ii) The properties of the resistance metric rely on the properties of the underlying Laplacian. It is shown in Proposition 2.6 that if Deg(x) ≤ Deg max < ∞ for all x ∈ V, we have that ρ is intrinsic with (iii) If ρ is intrinsic and all ρ-balls are nite, then G is complete [20,Theorem 2.8]. For the reader's convenience, we recall that a graph G = (V , w, m) is complete in the sense of [20] if there exists a nondecreasing sequence of nitely supported functions The operator Γ not only leads to a de nition of a metric, but also to the curvature conditions in the sense of Bakry-Émery.
We will need di erent assumptions to guarantee the semigroup characterization of Bakry-Émery curvature (see [17,20]). These assumptions are satis ed whenever the vertex measure m is non-degenerate, that is, inf and all balls with respect to an intrinsic metric are nite. In the case of bounded vertex degree Deg(x) ≤ Deg max for all x ∈ V, the non-degenerate vertex measure condition can usually be dropped.
In case of bounded vertex degree, the combinatorial distance is intrinsic up to a constant. Furthermore, we have a uniform control of the dimension in terms of the curvature. Proof. Let x, x ∈ V and let g := ρ(x , ·) We have where the latter inequality follows from Cauchy-Schwarz. Hence, G satis es the condition CD(K − s, Deg max s ) as claimed.
The main theorem stated below extends Theorem 1.1 to the case of negatively curved vertices.
for all x ∈ V, and assuming, for K, K > , 1. If N < ∞, V ≠ ∅, and assuming, for K, K > , Note that (i) and (ii) in the above theorem are included in [28] since every intrinsic metric is dominated by the resistance metric and, therefore, diamρ(G) ≤ diamσ(G).
We also point out that any locally nite graph with Deg max < ∞ is complete by Proposition 2.6 and Remark 2.4 (iii).

CD conditions and semigroups
By the spectral calculus, we can associate to ∆ the heat semigroup (P t ) t≥ . Using a standard argument, we derive a commutation formula for the semigroup and the gradient depending on the set of negatively curved vertices.  It is well known that the heat semigroup is generated by a smooth integral kernel, which is called the heat kernel. In particular, it can be proved that there is a pointwise minimal version, called p : ( , ∞) × V × V → R, obtained via an exhaustion procedure by Dirichlet heat kernels on compact (= nite) subsets of V (see, e.g., [26,39]).Therefore, we get Therefore, we have Rearranging yields the claim.
To control the distance to the negatively curved part V , we need to estimate P t V (x) in terms of ρ(x, V ). This is given by the following theorem.
Proof. From (3.2), we have for any bounded function g with bounded Γg, Hence, . Then, Γg ≤ /R and g + W ≤ and thus by (3.4), This nishes the proof.
We show that a Bonnet-Myers type diameter bound still holds if one allows some negative curvature. In contrast to Bonnet-Myers, we will bound the distance to the negatively curved part V of the graph from above, which proves part (iv) of Theorem 2.7.
On V \ T ρ R (V ), we have ρ(V , ·) ≥ R. Activating (3.5) towards |∆P T f | = |∂ T P T f | and throwing away the nonnegative term ΓP T f yields On the other hand, activating (3.5) towards ΓP T f and throwing away the nonnegative term . This gives good control on the time derivative and gradient of the semigroup. We estimate Moreover, for t < T, one has By assumption, one has ρ(x, y) ≤ Rρ whenever x ∼ y. We x T, R, r > and x ∈ V. We suppose ρ(x , V ) = R + Rρ + r. Our aim is to show that Let us explain the strategy of the remaining proof rst. We will consider functions f with Γf ≤ and being constant outside of Br(x ). We need the additional distance R to have reasonable estimates for ∂P t f and ΓP t f for all vertices in B r+Rρ (x ). The Rρ is needed to separate Br(x ) and T ρ R (V ), i.e., to guarantee that there are no edges connecting two vertices from the two sets respectively.
The distance R will be chosen later to ensure that the term We write Cmax := sup{f (y) : f ∈ F, y ∈ V} < ∞ due to connectedness.
The idea is to take f ∈ F such that sup f is (close to be) maximal. Then, we take P t f and cut it o appropriately such that its cut-o version h belongs to F. By the estimate (3.6) for |∂ t P t f |, we can upper bound |P t f − f | outside the tube T ρ R (V ). On the other hand for y ∉ Br(x ), we can upper bound |P t f (x ) − P t f (y )| by the estimate (3.7) for ΓP t f . By triangle inequality, we can thus upper bound (the cut-o version of) Notice that |f (y )−f (x )| ≈ Cmax ≥ r, this leads to an upper estimate for r when choosing R and T appropriately.
The reason, why we take Cmax as a substitute for the distance ρ, is that we need to forth-and back estimate between the distance and the gradient. The problem is that for ρ, we do not always have f (x) − f (y) ≤ ρ(x, y) when only assuming Γf ≤ . To avoid this problem, we take a certain resistance metric between x and V \ Br(x ) given by Cmax.
We now give the details. Let ε > . We choose f ∈ F such that C := sup f ≥ Cmax − ε. W.l.o.g., we can assume that f ≥ . We have Cmax ≥ r since the functionf := min(ρ(x , ·), r) ∈ F and supf = r . Now, we set gmax := inf{P T f (y) : y ∈ B r+Rρ (x ) \ Br(x )} and where a ∧ b := min{a, b} for a, b ∈ R. We remark that B r+Rρ (x ) \ Br(x ) ≠ ∅ due to the Rρ assumption (that is, ρ(x, y) ≤ Rρ for x ∼ y) and since V ≠ ∅ and since there is a path from x to V due to connectedness. The reason why we take the in mum over B r+Rρ (x ) \ Br(x ) and not over V \ Br(x ) is that we want to control |P T f (y ) − f (y )| at y where the in mum of P T f (·) is almost attained. But this only works if y is far away from the negatively curved set V .
In fact, we have (g(y) − g(z)) ≤ (P T f (y) − P T f (z)) , for all y, z ∈ V . (3.10) We can check (3.10) as follows. When y, z ∈ Br(x ), we have When one of the two vertices y and z lies in Br(x ) and the other one lies outside Br(x ), say y ∈ Br(x ) and z ∈ V \ Br(x ), we have (g(y) − g(z)) = (P T f (y) ∧ gmax − gmax) .
In case that P T f (y) ≤ gmax, we have by the de nition of gmax that Otherwise when P T f (y) > gmax, we have When y, z ∈ V \ Br(x ), we have (g(y) − g(z)) = (gmax − gmax) = ≤ (P T f (y) − P T f (z)) .
We observe that for all y ∈ V \ B r+Rρ (x ), there is no neighbor of y in Br(x ) due to the Rρ assumption. Since g is constant on V \ Br(x ), we obtain Γg = on V \ B r+Rρ (x ). Using ( . ), we derive further that where we used the property Γf ≤ . We will later choose T and R such that this bound of Γg is signi cantly smaller than one. Setting the function h : V → R to be Let y be a vertex in B r+Rρ (x ) \ Br(x ) such that P T f (y ) − g(y ) < ε. We obtain by (3.6) Analogously, we have Noticing that f (y ) = sup f = C and putting together (3.11), (3.12) and (3.13) yield Letting ε tend to zero yields whenever the denominator is positive. We set T := /K and R := √ NH(K, K , /K). Then the denominator of the RHS of (3.14) is − e − + / > . Observe that (3.8) implies Next, we estimate the numerator. By (3.9), we have for t ≤ T = /K that Therefore, we obtain Thus, (3.14) implies that It is left to show that e K /K N K +K is the dominating term in the sum and to give the corresponding coe cient.
We start with comparing the addends in the brackets of (3.17): we have and, hence, Notice that one has for s ≥ , e s ≥ √ + s.
This nishes the proof.
Combining Theorem 3.4 and Proposition 2.6, we obtain a distance bound for bounded vertex degree and in nite dimension, what proves part (iii) of Theorem 2.7. Corollary 3.6. Let G = (V , w, m) be a graph with nite maximal vertex degree Deg max , and let K, K > . Let ∅ ≠ V ⊂ V and suppose that G satis es Then, for all x ∈ V, one has d(x, V ) ≤ + e K /K Deg max KK .
Proof This nishes the proof.