Ricci curvature on polyhedral surfaces via optimal transportation
Abstract
The problem of defining correctly geometric objects such as the curvature is
a hard one in discrete geometry. In 2009, Ollivier [Oll09] defined
a notion of curvature applicable to a wide category of measured
metric spaces, in particular to graphs. He named it coarse Ricci curvature
because it coincides, up to some given factor, with the classical Ricci
curvature, when the space is a smooth manifold. Lin, Lu & Yau [LLY11],
Jost & Liu [JL11] have used and extended this notion for graphs giving
estimates for the curvature and hence the diameter, in terms of the combinatorics.
In this paper, we describe a method for computing the coarse Ricci curvature
and give sharper results, in the specific but crucial case of polyhedral surfaces.
Keywords: discrete curvature; optimal transportation; graph theory; discrete laplacian; tiling.
MSC: 05C10, 68U05, 90B06
1 The coarse Ricci curvature of Ollivier
Let us first recall the definition of the coarse Ricci curvature as given originally by Ollivier in [Oll09]. Since our focus is on polyhedral objects, we will use, for greater legibility, the language of graphs and matrices, rather than more general measure theoretic formulations. In this section, we will assume that our base space is a simple graph (no loops, no multiple edges between vertices), unoriented and locally finite (vertices at finite distance are in finite number) for the standard distance (the length of the shortest chain between and ):
where stands for the adjacency relation and adjacent vertices are
at distance . We shall call this distance the uniform distance; the more
general case will be considered in section 4.
Polyhedral surfaces (a.k.a. twodimensional cell complexes) studied afterwards will be seen
as a special case of graphs. We do not require our graphs to be finite, but we will
nevertheless use vector and matrix notations (with possibly infinitely many
indices).
For any , a probability measure on is a map such that . Assume that for any vertex we are given a probability measure . Intuitively is the probability of jumping from to in a random walk. For instance we can take to be the uniform measure on the sphere (or 1ring) at , , namely if , where denotes the degree of , and elsewhere. A coupling or transference plan between and is a measure on whose marginals are respectively:
Intuitively a coupling is a plan for transporting a mass distributed according to to the same mass distributed according to . Therefore indicates which quantity is taken from and sent to . Because mass is nonnegative, one can only take from the quantity , no more no less, and the same holds at the destination point, ruled by . We may view measures as vectors indexed by and couplings as matrices, and we will use that point of view later.
The cost of a given coupling is
where cost is induced by the distance traveled. The Wasserstein distance between probability measures is
where the infimum is taken over all couplings between and (such a set will never be empty and we will show that its infimum is attained later on). Let us focus on two simple but important examples:

Let be the Dirac measure , i.e. equals when and zero elsewhere. Then there is only one coupling between and and it satisfies if and , and vanishes elsewhere. Obviously .

Consider now the uniform measures on unit spheres around and respectively; then a coupling vanishes on whenever lies outside the sphere or outside . So the (a priori infinite) matrix has at most nonzero terms, and we can focus on the submatrix whose lines sum to and columns to . For instance, could be the uniform coupling
where we have written only the submatrix.

A variant from the above measure is the measure uniform on the ball .
Ollivier’s coarse Ricci curvature^{1}^{1}1Also called Wasserstein curvature. between and (which by the way need not be neighbors) measures the ratio between the Wasserstein distance and the distance. Precisely, we set
Since is the uniform measure on the sphere, then compares the average distance between the spheres and with the distance between their centers, which indeed depends on the Ricci curvature in the smooth case (see [Oll09, Oll10] for the analogy with riemannian manifolds which prompted this definition).
We will rather use the definition of Lin, Lu & Yau [LLY11] (see also Ollivier [Oll10]) for a smooth time variable : let be the lazy random walk
so that interpolates linearly between the Dirac measure and the uniform measure on the sphere^{2}^{2}2In [LLY11], a different notation is used: the lazy random walk is parametrized by and the limit point corresponds to .. We let , and as . We then set
and we will call the (asymptotic) Ollivier–Ricci curvature. The curvature is attached to a continuous Markov process whereas corresponds to a timediscrete process^{3}^{3}3However both approaches are equivalent, by considering weighted graphs and allowing loops (i.e. weights ). See [BJL11] and also §4 for weighted graphs.. Lin, Lu & Yau [LLY11] prove the existence of the limit using concavity properties. In the next section, we give a different proof by linking the existence to a linear programming problem with convexity properties.
The relevance of such a definition comes from the analogy with riemannian manifolds but can also be seen through its applications, e.g. the existence of an upper bound for the diameter of depending on (see Myers’ theorem below).
2 A linear programming problem
In the case of graphs, the computation of is surprisingly simple to understand and implement numerically. Recall that a coupling between and is completely determined by a submatrix, and henceforward we will identify with this submatrix. A coupling is actually any matrix in with nonnegative coefficients, subject to the following dependent linear constraints: and , for all and , where and are the following matrices
is the standard inner product between
matrices. We will write the nonnegativity constraint , where is the basis matrix whose coefficients
all vanish except at .
The set of possible couplings is therefore a bounded convex
polyhedron contained in the unit cube .
In the following, we will also need the limit set which contains a unique coupling (see case 1 above).
In order to compute , we want to minimize the cost function , which is actually linear:
where stands for the distance matrix restricted to , so that is the (constant) gradient of . Clearly the infimum is reached, and minimizers lie on the boundary of . Then either the gradient is perpendicular to some facet of , and the minimizer can be freely chosen on that facet, or not, and the minimizer is unique and lies on a vertex of . Moreover the Kuhn–Tucker theorem gives a characterization of minimizers in terms of Lagrange multipliers (a.k.a. Kuhn–Tucker vectors) : minimizes on if and only if there exists , and such that
(1) 
and
(2) 
meaning that the Lagrange multipliers have to vanish unless the inequality constraint is active (or saturated): . As a consequence, (i) finding a minimizer is practically easy thanks to numerous linear programming algorithms and (ii) proving rigorously that a given is a minimizer requires only writing the relations (1) and (2) for .
The nonuniqueness is quite specific to the metric, when cost is
proportional to length and therefore linear instead of strictly convex
(instead of, say, length squared as in the 2Wasserstein metric ).
It corresponds to the following geometric fact: transporting mass from to
is equivalent in cost to transporting the same mass from to
and from to , as long as is on a geodesic from to (and , since we prohibit negative mass).
Computing the Olivier–Ricci curvature requires a priori taking a derivative, however it is actually much simpler due to the following lemma, which also proves its existence, without the need for subtler considerations like in [LLY11]:
Lemma 1.
For small enough, convex sets and are homothetic. More precisely,
Proof.
First write the constraint corresponding to the lazy random walk as , where and iff . Let lie in and be positive. Then . Indeed lies in
We see immediately that whenever . Moreover . All the previous arguments hold in generality, but the nonnegativity of needs a different argument when . Because and are probability measures, , and for any , , provided
consequently . Hence, for , is positive for any matrix satisfying the equality constraints (and the same holds for , using again that ).
The signification of this positivity is that the constraint is never saturated: there will always be some mass transported from to if is small enough, because the other vertices cannot hold all the mass from . ∎
Remark 1.
The lemma holds true for small enough, as long as is uniformly bounded on , a property which we will meet later. More precisely if , then the homothety property holds for all .
Proposition 2.
The Ollivier–Ricci curvature is equal to any quotient for small enough (e.g. ).
Proof.
As a consequence of lemma 1, the gradient has the same projection on the affine space determined by the equality constraints, and the minimizers can be chose to be homothetic for small enough. If denotes this family of homothetic minimizers:
So is linear for small enough and
As a consequence, computing is quite simple: one needs only solve the linear problem for small enough (e.g. ).
Remark 2.
This property linking the timecontinuous Olivier–Ricci curvature to the timediscrete curvature is true in generality, as soon as the random walk is lazy enough, i.e. the probability of staying at is large enough (see [Vey12]).
Finally, we note that this optimization problem is an instance of integer linear programming and as a consequence, the solution is integervalued up to a multiplicative constant:
Theorem 3.
For any pair of adjacent vertices with degrees , and , there exists an optimal coupling with coefficients in ; consequently and lie in .
Proof.
Let us first rewrite the constraints above as the following single linear equation. Numbering and the neighbours of , and the neighbours of , we consider the vector
The constraints amounts to for the following data
The integral matrix is totally unimodular: Every square, nonsingular submatrix of has determinant . Indeed satisfies the following requirements:

the entries of lie in

has no more than 2 nonzero entries on each column

its rows can be partitioned into two sets and such that if a column has two entries of the same sign, their rows are in different sets.
Then, whenever is integervalued, the vertices of the constraint set are also integervalued. We refer the reader to classical results of integer linear programming which can be found in [PS98].
In our setting, choose , so that the coefficients of lie in . By the above remarks, so do the coefficients of , since an optimal coupling can be chosen to be a vertex of the contraint set. Since the distance matrix in also integervalued, the cost lies in , and for two neighbors , . The curvature is obtained by diving by , hence the result. The reasoning also holds for . ∎
3 Curvature of discrete surfaces
Estimates for the Ollivier–Ricci curvature are given in [LLY11] and [JL11] ( in the first paper, in the second) for general graphs and for some specific ones such as trees. Essentially they rely on studying one coupling, which gives an upper bound on , hence a lower bound on the curvature, which may or may not be optimal. We will give below exact values, albeit in the specific setting which concerns us: polyhedral surfaces. Furthermore, we will always assume that vertices are neighbors; in other words, we see as a function on the edges. Actual computing of for more distant vertices is of course possible, but much more complicated. However, it should be noted that trivially enjoys a concavity property, as a direct consequence of the triangle inequality on the distance : if is a geodesic path from to then
(3) 
the latter equality holding only in the uniform metric, because between neighbors. This inequality passes to the limit and applies to as well. The concavity property implies in particular that if is bounded below on all edges, then has the same lower bound on all couples .
We use this fact to give a trivial proof of Myers’ theorem (see also [GHL90] for the smooth case).
Theorem 4 (Ollivier [Oll09, prop. 23]).
If is bounded below on all edges by a positive constant , then is finite, and its diameter is bounded above by .
Proof.
Using the triangle inequality again on
where is the jump at , which is also the expectation of the distance to w.r.t. the probability . For the uniform metric , so that
which gives the upper bound for the diameter. Since is locally finite, it is therefore finite. ∎
We will now give our results, and compare them with those obtained either by Jost & Liu [JL11] or by using Forman’s definitions of Ricci curvature [For03].
As first example, let us give the Ollivier–Ricci curvature for the Platonic solids (with as a comparison, corresponding to the nonlazy random walk) in table 1:
tetrahedron  cube  octahedron  dodecahedron  icosahedron  
Forman 
This stresses the difference between (used in [JL11]) and , which exhibits, in our opinion, a more geometric^{4}^{4}4And less graphtheoretic. behavior. In particular, the values of are sharp w.r.t. Myers’ theorem for the cube and the octahedron. Forman refers to the combinatorial Ricci curvature for unit weights defined in [For03], which also satisfies a Myers’ theorem, albeit with a different constant: the diameter is bounded above by , hence our choice to divide it by , to allow comparison between with the Ollivier–Ricci curvature. Here, only the cube is optimal.
Tessellations by regular polygons fit well in this framework since all edges have the same length. Regular tiling are the triangular, square and hexagonal tiling. The triangular tiling corresponds to the case above and has zero Ollivier–Ricci curvature, and so does the square tiling. However the hexagonal lattice has negative Ollivier–Ricci curvature equal to .
The method can also be applied to semiregular tiling, but those are only vertextransitive in general
and not edgetransitive (with the exception of the trihexagonal tiling),
hence one must treat separately the different types of edges. For example, for the snub square tiling,
for an edge between two triangles, but for an edge between
a triangle and a square.
The results above can easily be derived using making computation by hand or by using integer linear programming software (a program with all the above examples using opensource software Sage^{5}^{5}5http://www.sagemath.org/ is attached to the article). The next results however are of a more general nature, with variable degrees, and cannot be obtained by simple computations. We consider adjacent vertices on a triangulated surface with the following genericity hypotheses:
 (B)

are not on the boundary,
 (G)

for any and , there is a geodesic of length in .
Under hypothesis (G), the distance matrix in agrees with its restriction to , hence all computations are local. For this genericity assumption to fail, one needs very small loops close to and , which can usually be excluded as soon as the triangulation is fine enough. Note that the Platonic solids are not generic in that sense, and many other configurations are ruled out (e.g. ). Then we conclude with the following.
Theorem 5.
Under the genericity hypotheses (B) and (G), the Ollivier–Ricci curvature depends only on the degrees of vertices and is given in table 2.
Proof.
To compute the optimal cost , we need only find a coupling for which the KuhnTucker relation (1) holds. Thanks to the genericity hypothesis (G), we can restrict ourselves to finite matrices (on ). Details are given in the section 5. Note that the hypothesis (B) makes for simpler calculations, but they could obviously be extended to deal with the presence of boundary. ∎
The table 2 gives and in function of respective degrees . Because we may assume without loss of generality that . We compare with Forman’s expression and also to the lower bound
given by Jost & Liu [JL11] for general graphs, where is the number of triangles incident to the edge (), which under our hypotheses is always equal to . Jost & Liu conclude that the presence of triangles improves the lower Ricci bound. We see here that when there are only triangles one obtains an actual value, which differs from their lower bound as soon as .
Remark 3.

The case is given here although it contradicts either (B) or (G), the latter being the tetrahedron computed above; similarly the case is excluded.

Zero Ollivier–Ricci curvature is attained only with degrees (regular triangular tiling), and .
4 Varying edge lengths
While many authors have focused on the graph theory, the case of polyhedral
surfaces is somewhat different: The combinatorial structure is more
restrictive, as we have seen above, but the geometry is more varied. In
particular, edge lengths may be different from one.
This is partially achieved in the literature [LLY11, JL11]
by allowing weights on the edges, which amounts to changing the random walk,
but we think the geometry should intervene at two levels: measure and distance.
We will present here a general framework to approach the problem, using the Laplace operator,
which depends on both the geometric and the combinatorial structure of .
One must also note the ambiguous definition of the Ollivier–Ricci asymptotic curvature,
which plays the role of a length in Myers’ theorem, and yet its definition
makes it a dimensionless quantity. Indeed multiplying all lengths by a constant
will not change ( being multiplied by as well).
In the following we assume that is a polyhedral (or discrete)
surface with set of vertices , edges and faces . Furthermore
is not only locally finite, but its vertices have a maximum degree
( denotes the minimal degree, which is at least for
surfaces with boundary, and for surfaces without boundary). The geometry
of is determined by the geometry of its faces, namely a isometric
bijection between each face and a planar face of identical degree, with
the compatibility condition that edge lengths measured in two adjacent faces
coincide. Then two natural notions of length arise: (i) the combinatorial
length, which counts the number of edges along a path and (ii) the metric
length, where each edge length is given by the geometry. Each notion of length
yields a different distance between vertices: the combinatorial distance
, which we have used above, and the metric distance . Note that if
each face is assumed to be a regular polygon with edges of length one, then
both distances agree, and metric theory coincides with graph theory. We will
make the following assumption on the geometry: the distance and
are metrically equivalent: . Such an hypothesis holds if the lengths of edges are
uniformly bounded above and below; in particular, the aspect ratio is
bounded^{6}^{6}6This also rules out extremely large or extremely
small faces, which could happen with only the bounded aspect ratio..
We consider a differential operator (a laplacian, see [CdV98]) determined by its values for vertices and the usual properties^{7}^{7}7Note that our sign convention is such that the Laplacian is a negative operator; [CdV98] uses the opposite.:

whenever ,

whenever and (locality property)

, which implies that (note that the sum is finite due the previous assumption and the local finiteness of ).
Often this operator is obtained by putting a weight on each edge . The degree at is then the sum and . Obviously property (c) implies . The case studied above corresponds to a graph with all weights equal to one (therefore unweighted), and the corresponding Laplace operator is called the harmonic laplacian .
The laplacian is not a priori symmetric, i.e. selfadjoint (though it could be made so w.r.t. some metric on vertices). Thanks to the finiteness assumption (b), we can define iterates of for integer , and the coefficient (not to be confused with ) is
the sum being taken on all paths of length on . By direct recurrence, we see that our boundedness hypotheses imply the bound . Indeed,
As a consequence, the heat semigroup is welldefined. It acts on measures, and defines the image measure of the Dirac measure at by
where is the lazy random walk studied above (for the harmonic laplacian, but results hold in the general case). The random walks have finite first moment, as can be inferred from the proof of the following.
Proposition 6.
The Ollivier–Ricci curvature depends only on the first order expansion of the random walk:
Proof.
Consider any coupling that transfers mass from points at (uniform) distance from at least 2, to and its neighbors. If the vertex is at distance from , then for and
The points at uniform distance from are at most numerous, and using the equivalence between distances, they will be moved at most by to or one of its neighbors:
Since
we conclude that both limits coincide. ∎
As a consequence, it is natural to replace in the section above the random walk by , for some definition of the Laplacian (see [BS07, WMKG08, AW11]). However in order to recover the geometric properties above one needs to normalize the random walk , so that the jump , i.e. the average distance of points jumping from should be . That amounts to setting:
equivalently one might renormalize the laplacian accordingly. As a consequence,
now behaves as the inverse of a length, as expected. Furthermore,
Myers’ theorem 4 is still valid. Indeed, while equation (3)
no longer holds when edge lengths vary, it remains true that if is
bounded below on all edges by .
An example: The rectangular parallelepiped.
For the rectangular parallelepiped with edges of lengths , the Ollivier–Ricci curvature is
along an edge of length , and others follow (see §5.4). For the cube, we recover . If is the length of the longest edge, an application of Myers’ theorem yields an upper bound for the diameter times greater than its actual value .
Remark 4.
A more general theory can be developed with nonlocal operators, by replacing local finiteness (property (b) above) with convergence requirements. Another, still finite, natural generalization of (b) is to allow whenever and belong to the same face. For a triangulated manifold this amounts to the usual neighborhood relation, but as soon as some faces have more than three edges, this makes a difference (e.g. the cube). Note however that the corresponding Myers’ theorem needs to be adjusted as well since the jump will change accordingly. In our experiments on Platonic solids with a uniform measure on vertices of , we did not find better diameter bounds with this method.
Remark 5.
One might also be tempted to compute Ollivier–Ricci curvature on the surface seen as a smooth flat surface with conical singularities (so that distances are computed between points on the faces). If vertices both have nonnegative Gaussian curvature (a.k.a. angular defect ) then by a computation analog to Ollivier’s [Oll09], we infer
which differs from our previous computations. This emphasizes that this setup is somewhere in between the smooth and the discrete setup.
5 Appendix: solutions for the linear programming problem on generic triangulated surfaces
We give here the Lagrange multipliers for the linear programming problem and the corresponding minimizer. The regular tetrahedron is given first as an example of the method, and the main result consists of analyzing the various cases according to their (arbitrary) degrees. Cases with degrees less or equal to can easily be computed by a machine and we refer to the Sage program attached.
5.1 The regular tetrahedron
The distance matrix for vertices is
and the optimal coupling from to shifts mass from vertex to vertex (provided ), leaving other vertices untouched:
with Lagrange multipliers
the last matrix corresponding to a linear combination of with positive coefficients , only where . Conversely, it is straightforward from to deduce that is unique. Hence and . The case cannot be dealt with in the same way, but admits the following optimal transference plan
with cost and therefore curvature .
Remark 6.
The case of degrees differs only in that the distance between vertices and is equal to instead of . However the optimal couplings found above do not move mass from nor from . Hence it is also optimal for the case.
5.2 Generic triangulated surfaces
We analyse now generic triangulated surfaces according to the degrees of and . In our matrix notation, will have index and index 2. Since and are not on the boundary, all edges containing them belong to two triangular faces. In particular there are two vertices, with indices and , that are neighbors of both and (see figure 1). There remains exclusive neighbors of (that are not neighbors of ), ordered from to along the border of , and exclusive neighbors of , ordered from to along the border of .