Long scale Ollivier-Ricci curvature of graphs

We study the long scale Ollivier-Ricci curvature of graphs as a function of the chosen idleness. As in the previous work on the short scale, we show that this idleness function is concave and piecewise linear with at most $3$ linear parts. We provide bounds on the length of the first and last linear pieces. We also study the long scale curvature inside the Cartesian product of two regular graphs.


Introduction and statement of results
Ricci curvature is a fundamental notion in the study of Riemannian manifolds. This notion has been generalized in various ways from the smooth setting of manifolds to more general metric spaces. For example, in [11] Ollivier introduced a notion of Ricci curvature on metric spaces (later known as "Ollivier Ricci curvature"). This gives rise to a notion of Ricci curvature on graphs taking values on the edges and based on optimal transport of lazy random walks, with respect to an idleness parameter p. In [6] this notion was modi ed on graphs to give the "Lin-Lu-Yau" curvature.
In [1] the authors investigate the Ollivier Ricci idleness function p → κp(x, y), which takes the idleness parameter p ∈ [ , ] as a variable and gives the value of curvature between the xed two adjacent vertices x and y (or equivalently, the curvature given on an edge of the graph joining x and y). End this sentence with a full-stop, after the intervals expression p → κp(x, y) is concave and piecewise linear on [ , ] with at most 3 linear parts, and it is linear on the intervals , lcm(dx , dy) + and max(dx , dy) + , In this paper, we do similar investigation on the idleness function, but the condition that the two vertices are adjacent is replaced by distance ≥ apart (and henceforth called "long-scale curvature" as in contrast to "short-scale curvature"). Our main result is that the (long-scale) idleness function p → κp(x, y) is concave and piecewise linear on [ , ] with at most 3 linear parts, and it is linear on the intervals , lcm(dx , dy) + and − · dx + dy dx dy − dx − dy , .
In a speci c case when dx = or dy = , the idleness function p → κp(x, y) is linear on the entire interval [ , ]. This main result is split into two theorems, which are stated and proved in Section 3 and 4. In Section 5, we provide an example of a graph that has exactly 3 linear parts and the rst and the last linear parts are the same intervals as mentioned in the main result. In Section 6, we give the formula of the long-scale curvature of Cartesian products of regular graphs. In Section 7, we present some interesting behaviours of the long-scale curvature, including the hexagonal tiling, and the discrete Bonnet-Myers' theorem.

De nitions and notation
We now introduce the relevant de nitions and notation we will need in this paper.
Throughout this article, let G = (V , E) be a simple graph (i.e., G contains no multiple edges or self loops) with a vertex set V and an edge set E. Furthermore, we assume that G is locally nite and connected. Let dx ∈ N denote the degree of the vertex x ∈ V and d(x, y) ∈ N ∪ { } denote the combinatorial distance, that is, the length of a shortest path (also called a geodesic) between two vertices x and y. We also denote the existence of an edge between x and y by x ∼ y.
A probability measure µ on V is a function µ : V → [ , ] satisfying v∈V µ(v) = . All probability measures are assumed to be nitely supported, that is is a nite set. For any x ∈ V and p ∈ [ , ], the probability measure µ p x is de ned as A -sphere and a -ball around a vertex x ∈ V are de ned as In particular, supp(µ p x ) ⊆ B (x) for all p ∈ [ , ]. Let W denote the 1-Wasserstein distance between two probability measures (see [14, pp. 211]). Its de nition on graphs can be written as follows.
De nition 2.1. Let G = (V , E) be a locally nite and connected graph. Let µ , µ be two probability measures on V. The Wasserstein distance W (µ , µ ) between µ and µ is de ned as A function π ∈ (µ , µ ) is called a transport plan transporting µ to µ (and π is said to have marginals µ and µ ). In words, a transport plan π moves a mass distribution given by µ into a mass distribution given by µ , and W (µ , µ ) is a measure for the minimal e ort which is required for such a transition. If π attains the in mum in (2.1) we call it an optimal transport plan transporting µ to µ .
De nition 2.2. Let G = (V , E) be a locally nite and connected graph. For p ∈ [ , ], the p−Ollivier Ricci curvature of two di erent vertices x, y ∈ V is where p ∈ [ , ] is called the idleness parameter. Moreover, the Lin-Lu-Yau curvature is In particular, we call the curvature κp(x, y) and κ LLY (x, y) "short-scale" when x ∼ y, and we call it "longscale" when d(x, y) ≥ .
A fundamental concept in optimal transport theory and vital to our work is Kantorovich duality. First we recall the notion of 1-Lipschitz functions and then state the Kantorovich duality theorem.

De nition 2.3. Let G = (V , E) be a locally nite and connected graph. For any
Furthermore, let 1-Lip denote the set of all -Lipschitz functions on V. Theorem 2.1 (Kantorovich duality [14]). Let G = (V , E) be a locally nite and connected graph. Let µ , µ be two probability measures on V. Then If ϕ ∈ 1-Lip attains the supremum we call it an optimal Kantorovich potential transporting µ to µ .
Remark 2.2 (Existence of optimal transport plans and optimal Kantorovich potentials). The de ning equation (2.2) can be realized as a nite-dimensional linear minimization problem on the bounded convex set (µ , µ ), so this problem admits a minimizer π. Furthermore, its dual problem can be written as It is well-known that any -Lipschitz function on U ⊆ V can always be extended to a -Lipschitz function on V (see [8]). In particular, a maximizer Φ is extended to a maximizer ϕ in the equation (2.3). This argument guarantees the existence of an optimal transport plan and an optimal Kantorovich potential transporting µ to µ .
We now present the complementary slackness theorem in the setting of our optimal transport problem, which says that the -Lipschitz condition on any optimal Kantorovich potential holds with equality on the support of any optimal transport plan. A similar statement and a proof can be found in [1, Lemma 3.1].

Proposition 2.5 (Integer-Valuedness).
Let G = (V , E) be a locally nite and connected graph. Let µ , µ be two probability measures on V. Then there exists an integer-valued optimal Kantorovich potential ϕ : V → Z transporting µ to µ , that is (
to be the fractional part of ϕ * (v). Let π be an optimal transport plan transporting µ to µ , and denote U = supp(µ ) ∪ supp(µ ). Construct a graph H with vertices in U and edges given by its adjacency matrix A H : Since U is nite, we may denote the connected components of H as which implies that δv − δw has an integer value. Since δv − δw ∈ (− , ), it must be . In conclusion, δv = δw for all v H ∼ w. By transitivity, δv is constant throughout all vertices v in the connected component U i , for each i. We may then de ne δ i := δv for any v ∈ U i . Now observe that because for each v ∈ U i and each w ∈ supp(µ ) such that π(v, w) > , we know that v H ∼ w, so w ∈ U i . By similar arguments, Therefore ϕ * is an optimal Kantorovich potential as desired.

The idleness function is 3-piece linear
In this section, we will prove one of the main results: for any x, y ∈ V such that d(x, y) ≥ , the idleness function p → κp(x, y) is piecewise linear with at most 3 linear parts. The proof follows the method from Theorem 3.4 in [1], which proves the result in case x ∼ y. First, we need the following lemma. Proof. Let π * ∈ (µ p x , µ p y ) and ϕ * : V → R be an optimal plan and an optimal Kantorovich potential transporting µ p x to µ p y . The marginal constraints of π * give v∈B (x) π * (v, y) = µ p y (y) = p > , which implies that π * (x , y) > for some x ∈ B (x). By the complementary slackness theorem, ϕ * (x ) − ϕ * (y) = d(x , y). By the Lipschitz and metric properties, we then have Subtracting the rightmost term from the leftmost term gives π * (x, y) > , and the complementary slackness theorem implies that ϕ * (x) − ϕ * (y) = d(x, y) = δ.

Theorem 3.2. Let G = (V , E) be a locally nite and connected graph, and let x, y ∈ V with d(x, y) = δ ≥ . Then p → κp(x, y) is concave and piecewise linear over [ , ] with at most 3 linear parts.
Proof.
and for j ∈ {δ − , δ − , δ}, de ne a set Moreover, de ne a constant c j := sup and therefore Hence, p → κp(x, y) is concave and piecewise linear with at most 3 linear parts.

Remark 3.3.
For p > , in the second line of equations (3.1), the condition on the supremum can be replaced by ϕ(x)−ϕ(y) = δ, due to the second half of Lemma 3.1. Doing so gives W (µ p x , µ p y ) = f δ (p) for all p > . In other words, the idleness function p → κp(x, y) has the last linear part (at least) on the interval [ , ]. The same statement is also true in case x ∼ y (see [1], Theorem 4.4). One immediate consequence is the simpli cation of the Lin-Lu-Yau curvature.

Critical points of the idleness function
In this section, we will discuss about the length of each linear part of the idleness function in terms of "critical points".

De nition 4.1 (critical points). De ne critical points (of κp(x, y)) to be the values
In other words, critical points are the values of p where the changes of slopes of the function p → κp(x, y) happen. We may replace κp(x, y) by W (µ p x , µ p y ) because they are closely related by the linear relation: so they share the same critical points. Since the idleness function has at most 3 linear pieces, there are at most two critical points. Our goal of this section is to determine the possible values of the critical points.
The following proposition shows that, in case of dx = or dy = , the idleness function is actually linear on the entire interval [ , ], so there is no critical point. As a consequence, all results about critical points will be discussed under the assumption that dx , dy ≥ . Proof. Without loss of generality, assume that dx = and let x ∼ x . Observe that every geodesic starting from x must pass through x . In other words, Consider an optimal transport plan π from µ p x to µ p y . The distance W (µ p x , µ p y ) can be derived as which is linear in p, so is κp(x, y). Here, the second line of the equation above uses the fact that B (x) = {x, x }, and the last line uses the marginal constraints of π: Next is the main theorem of this section, which gives an upper bound on the values of critical points. Such a bound is sharp, as shown and explained in the Section 5.
The key of the proof lies in the following two lemmas. The rst one compares the terms c j 's introduced in the proof of Theorem 3.2. The second one gives an explicit formula for critical points in terms of c j 's.

Lemma 4.3.
With the same setup as above, The proof of Lemma 4.3 is postponed towards the end of this section.

Lemma 4.4.
With the same setup as above, de ne constants p , p ∈ R to be Then, for all t ∈ R, the functions f j as de ned in Theorem 3.2 satisfy Proof of Lemma 4.4. First, note that the denominators of p and p are positive real numbers, due to Lemma 4.3. Next, we show that p ≤ p . Consider the function g : (− , ∞) → R de ned by which is an increasing function on t.
Next, we compare f δ− and f δ− . From the de nition f j (t) = t · j + ( − t)c j , we have Similarly, a comparison between f δ− and f δ gives: By the above comparisons, we can then conclude the equation Since ∂ ∂t f j = j − c j , it means that the slopes of f δ− ,f δ− , and f δ are all di erent: The second statement in the lemma immediately follows by renaming the variable t as p with a further restriction p ∈ [ , ].
Proof of Theorem 4.2. Recall the function g de ned in the proof of Lemma 4.4. The monotonicity of g together with Lemma 4.3 implies that which concludes the proof of the theorem.
Now we come back to prove Lemma 4.3.
Proof of Lemma 4.3. First, we prove the rightmost inequality: Consider ϕ * : V → Z such that ϕ * ∈ A δ− and F(ϕ * ) = sup  N and b , b , On the other hand, the Lipschitz property implies ϕ * (a) − ϕ We will now show that ϕ is -Lipschitz. It is su cient to show that ϕ (w) − ϕ (z) ≤ for any w, z ∈ V such that w ∼ z. By de nition of ϕ , we have which is less than or equal to , except when ϕ * (w)−ϕ * (z) = and 1 Vx (w) = and 1 Vx (z) = , simultaneously.
These exception conditions would imply that z is a child of w, and w x, and z ̸ x, which is impossible as it contradicts to the transitivity of partial ordering. Therefore, ϕ is -Lipschitz as desired. Moreover, ϕ ∈ A δ (because ϕ (x) = ϕ * (x) + = δ and ϕ (y) = ϕ * (y) = since y ∉ Vx).
Comparison between ϕ * and ϕ gives where S (x) and S (y) are the sets of neighbours of x and of y, respectively. A simple bound on (4.5) will give However, this inequality can be improved by the following 3-case separation.
• Case 1: It means that x has no child and hence no descendant, i.e. Vx = {x}. Thus • Case 3: S (y) ∩ Vx = S (y). It means that y ≺ x for all neighbours y of y. We now de ne a new function ϕ : V → Z by which is 1-Lipschitz and in A δ (similar as to how ϕ is 1-Lipschitz and in A δ ). It follows that Hence, c δ − c δ− ≥ .
From the three cases above, we can conclude the rightmost inequality in (4.2): Next, we use a similar method as above to prove the leftmost inequality:

De ne a set of vertices Vx ⊆ V by
Vx := {w ∈ V x w}.

Remark 4.5 (Length of the rst linear part)
. Note that each c j ∈ Z/l where l := lcm(dx , dy). Therefore, p and p must be in the form of a l a l + = a a+l for some a ∈ Z. Hence, the least possible value for a positive critical point is +l . In other words, rst linear part of the function p → κp(x, y) is at least the interval [ , +lcm(dx ,dy) ].

Remark 4.6 (Length of the last linear part). Theorem 4.2 says that the last linear part of the function
In a special case that vertices x and y have the same degree dx = dy = D ≥ , each critical point p * of κp(x, y) satis es Moreover, from the de nition, c j ∈ Z/D, so p , p must be in the form of a D+a for some integer ≤ a ≤ D − .
The next section provides an example of a graph with dx = dy = D where the inequality (4.7) holds with equality, that is, a critical point occurs exactly at D− D− .

An important family of examples
In this section we aim to construct a graph G = (V , E) with points x, y ∈ G such that d(x, y) ≥ and the idleness function p → κp(x, y) has three linear pieces and has one critical point as large as the one mentioned in (4.7). Let m, n, k be arbitrary natural numbers (including zero). De ne vertices of G to be and de ne edges of G to be If m, n or k is zero, we simply remove the related vertices and edges. The graph G is shown in Figure 1 in case m = n = k = (but the indexes m, n, k are kept in the labelling for clarity).
In the constructed graph G, we have d(x, y) = and dx = dy = D = + m + n + k. Our goal is to show that the function p → W (µ p x , µ p y ) has its critical points at m D+m and m+n D+m+n . In particular, if k = , then the larger critical point coincides with D− D− , the maximum possible value mentioned in Remark 4.6. We need to calculate the value of c j for each j ∈ { , , }. First, we start by giving a lower bound to c j by choosing an appropriate function ϕ j ∈ A j for each j.
De ne functions ϕ , ϕ , ϕ : V → Z as in Table 1. It can be easily checked that ϕ j ∈ A j for j ∈ { , , }, and hence we obtain the three following inequalities: Next, we give an upper bound to c , c , c by calculating the costs of transport plans π , π , π from µ p x to µ p y (with idleness p = , m D+m , m+n D+m+n , respectively). The plans π , π , π are constructed as in Table 2.
It is straightforward to check that π ∈ (µ x , µ y ), π ∈ (µ Lastly, Lemma 4.4 then gives a formula for W (µ p x , µ p y ) that where the critical points are

The Cartesian product
In [6] the authors proved the following results on the curvature of Cartesian products of graphs:  , y), (x , y) We now extend this result to the long-scale curvature.

Theorem 6.2. Let G = (V G , E G ) be a D G -regular connected graph and H = (V H , E H ) be a D H -regular connected
graph. Let x , x ∈ V G and y , y ∈ V H . Then (y , y )) .
Here we use the notation D G , D H , instead of d G , d H for the vertex degree to distinguish it from the distance function d(·, ·). Moreover, we use convention d(x , x )K G LLY (x , x ) = in case x = x , and d(y , y )K H LLY (y , y ) = in case y = y . Before proving the theorem, we introduce a lemma stating that the sum of 1-Lipschitz functions on two di erent graphs is a 1-Lipschitz function on the Cartesian product graph. Proof of Lemma 6.3. Let w , w ∈ V G and z , z ∈ V H . By applying 1-Lipschitz properties of ϕ G and ϕ H , we obtain yielding the lemma. The proof includes the four following steps: ( ) Show that the lower bound and upper bound of W (µ p (x ,y ) , µ p (x ,y ) ) given in ( ) and ( ) coincides for large enough p ∈ [ , ]. Hence, the inequality in Step (1) is indeed an equality for p large enough. ( ) Derive the Lin-Lu-Yau curvature on the Cartesian product.
Substitution q = λ and r = p λ gives where the second line uses the identity − λ = λ − p, yielding the equation (6.10).
Step (4) The previous steps imply that, for large enough p ∈ [ , ], Finally, we can translate this relation in term of the Lin-Lu-Yau curvature, using Corollary 3.4: for any a ≠ a ∈ A and any q ∈ [ , ], . (y , y )) .

Long-scale behaviour
In most papers regarding Ollivier Ricci curvature, only the short-scale curvature is usually considered because the curvature given at an edge x ∼ y is a discrete analogue to the Ricci curvature given at a unit tangent vector. Moreover, a lower bound on the short-scale curvature implies the same lower bound for the curvature between any two points (see [11,Proposition 19]): If κp(x, y) ≥ κ for all x ∼ y, then κp(x, y) ≥ κ for all x, y ∈ V .
However, restricting oneself only to the short-scale curvature could lead to some contradiction to the nature of particular graphs, e.g. the hexagonal tiling as illustrated in the following subsection. Later, we then discuss about some global implication of curvature signs.

. The hexagonal tiling
Let G = (V , E) be a graph of the hexagonal tiling (which may be either in nite tessellation, or nite tessellation, e.g. on a torus T ).
Consider a pair of points (x, y) with distance d(x, y) = . There are 4 non-equivalent positions of y relative to x listed as y , y , y , y as shown in Figure 2. The following proposition gives the formula of the short-scale and the long-scale curvature (of distance 7) in the hexagonal tiling. Sketch of proof. (a) Given w, z ∈ V with w ∼ z. Denote the neighbors of w (other than z) by w , w , and denote the neighbors of w (other than w) z , z . An optimal way to transport µ p w to µ p z is to transport the mass p from w to z, and transport the masses −p from w to w, from z to z , and from w to z . Then W (µ p w , µ p z ) = p · + − p ( + + ) = + ( − p), which implies κp(w, z) = − W (µ p w , µ p z ) = − ( − p).
(b) Given x, y with d(x, y) = in the hexagon tiling. There are two possible con gurations regarding the positions of the vertices in S (x) and S (y): either (b1) two corresponding vertices in S (x) and S (y) have distance , and another pair has distance , or (b2) two corresponding vertices in S (x) and S (y) have distance , and another pair has distance .
It can be checked that an optimal way to transport µ p x to µ p y is to transport the mass p from x to y, and transport the rest between the corresponding vertices mentioned above. This implies that W (µ p x , µ p y ) = p · + −p ( + + ) = − ( − p) for case (b1) p · + −p ( + + ) = + ( − p) for case (b2), so κp(x, y) = − W (µ p x , µ p y ) = ± ( − p). The construction of these optimal transport plans are illustrated by the diagrams in In particular, consider a nite hexagonal tessellation on a torus T , where the space is expected to have both positive and negative curvature. However, the short-scale curvature is negative everywhere on the hexagonal tessellation, which suggests that the long-scale curvature is more suitable to describe this space.

. Global results
Theorem 7.2 (non-positive curvature). Let G = (V , E) be a locally nite and connected graph and let p ∈ [ , ). Assume that the curvature κp(x, y) ≤ for all x ≠ y ∈ V. Then G must be in nite.
Proof. Suppose for the sake of contradiction that G is nite, with diameter diam(G) := sup{d(w, z) : w, z ∈ V} = L < ∞.
For a complete graph Kn, any edge x ∼ y satis es W (µ p x , µ p y ) = |p − −p n− |, so κp(x, y) > for every p ∈ [ , ). Since we assume G to be non-positively curved everywhere, G cannot be a complete graph, so L ≥ .
Let x and y be antipodal vertices in V, that is d(x, y) = L. Consider a geodesic from x to y, namely x = v ∼ v ∼ ... ∼ v L− ∼ v L = y. It follows that v is a neighbour of x, and v L− is a neighbour of y, and that d(v , v L− ) = L − . Consider a transport plan π ∈ (µ p x , µ p y ) such that π(v , v L− ) > .
Hence the W (µ p x , µ p y ) is bounded above by: W (µ p x , µ p y ) ≤ w,z∈V π(w, z)d(w, z) ≤ L · w,z∈V π(w, z) = L. (7.1) Moreover, π(w, z)d(w, z) < Lπ(w, z) when w = v and z = v L− , so the inequality in (7.1) must be strict. That is W (µ p x , µ p y ) < L which then implies κp(x, y) = − L W (µ p x , µ p y ) > , contradicting to the curvature assumption. In fact, this formula also holds in the case d(x, y) = . The in nite regular trees illustrate a family of graphs which have non-positive curvature everywhere.
Remark 7.4. The theorem of discrete Bonnet-Myers [6,11] states that a graph G = (V , E) with positive curvature bounded away from zero κp(x, y) ≥ K > for all x ≠ y ∈ V must be a nite graph. This assumption can be replaced by: κp(x, y) ≥ K > for all neighbours x ∼ y ∈ V, since both assumptions are essentially equivalent. On the other hand, the assumption in Theorem 7.2 cannot be reduced to: κp(x, y) ≤ for all neighbours x ∼ y ∈ V. As a counterexample, consider a graph G of a nite hexagonal tessellation on a torus T (see Subsection 7.1).
There is another way to modify discrete Bonnet-Myers' theorem, by replacing the assumption condition with κp(x, y) ≥ κ > for a xed vertex x ∈ V and for all y ∈ V\{x}. Theorem 7.5 (modi ed discrete Bonnet-Myers). Let G = (V , E) be a locally nite and connected graph and let p ∈ [ , ). Assume that there is a constant κ > and a xed vertex x ∈ V such that the curvature κp(x, y) ≥ κ for all y ∈ V\{x}. Then G must be nite.
The proof is very similar to the one in the original discrete Bonnet-Myers [11], which employs the Diracmeasure δx = 1x.