Ollivier’s Ricci Curvature, Local Clustering and CurvatureDimension Inequalities on Graphs
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Abstract
In this paper, we explore the relationship between one of the most elementary and important properties of graphs, the presence and relative frequency of triangles, and a combinatorial notion of Ricci curvature. We employ a definition of generalized Ricci curvature proposed by Ollivier in a general framework of Markov processes and metric spaces and applied in graph theory by Lin–Yau. In analogy with curvature notions in Riemannian geometry, we interpret this Ricci curvature as a control on the amount of overlap between neighborhoods of two neighboring vertices. It is therefore naturally related to the presence of triangles containing those vertices, or more precisely, the local clustering coefficient, that is, the relative proportion of connected neighbors among all the neighbors of a vertex. This suggests to derive lower Ricci curvature bounds on graphs in terms of such local clustering coefficients. We also study curvaturedimension inequalities on graphs, building upon previous work of several authors.
Keywords
Ollivier’s Ricci curvature Curvature dimension inequality Local clustering Graph Laplace operator1 Introduction
When one studies empirical graphs, one of the most obvious and basic properties to investigate is the presence and number of triangles, that is, connected triples of vertices. In bipartite graphs, for instance, there are no triangles, whereas in a complete graph, every triple of vertices constitutes a triangle. A basic observation then is that when two neighboring vertices are contained in a triangle, their neighborhoods of radius 1 (let us assign to every edge the length 1 for the discussion in this introduction) share the third vertex of the triangle. That is, the more triangles those two neighboring vertices are contained in, the larger the overlap of their neighborhoods. This suggests an analogy with the notion of Ricci curvature in Riemannian geometry where a lower bound on the Ricci curvature also controls the amount of overlaps of distance balls from below. This is what we are going to explore in a quantitative manner in this paper.
In fact, Ricci curvature is a fundamental concept in Riemannian geometry, see e.g. [18]. It is a quantity computed from second derivatives of the metric tensor. It controls how fast geodesics starting at the same point diverge on average. Equivalently, it controls how fast the volume of distance balls grows as a function of the radius. As already indicated, it also controls the amount of overlap of two distance balls in terms of their radii and the distance between their centers. In fact, such lower bounds follow from a lower bound on the Ricci curvature. It was then natural to look for generalizations of such phenomena on metric spaces more general than Riemannian manifolds. That is, the question to find substitutes for the lower bounds on the above mentioned second derivative combinations of the metric tensor that yield the same geometric control on a general metric space. By now, there exist several insightful definitions of synthetic Ricci curvature on general metric measure spaces, see Sturm [29, 30], Lott–Villani [22], Ohta [23], Ollivier [24] etc.
As indicated, in this paper, we want to explore the implications of such ideas in graph theory. The geometric idea is that a lower Ricci curvature bound prevents geodesics from diverging too fast and balls from growing too fast in volume. On a graph, the analogue of geodesics starting in different directions, but eventually approaching each other again, would be a triangle. Therefore, it is natural that the Ricci curvature on a graph should be related to the relative abundance of triangles. The latter is captured by the local clustering coefficient introduced by Watts–Strogatz [33]. Thus, the intuition of Ricci curvature on a graph should play with the relative frequency of triangles a vertex shares with its neighbors. In fact, more precisely, since the local clustering coefficient averages over the neighbors of a vertex, this should really related to some notion of scalar curvature, as an average of Ricci curvatures in different directions, that is, for different neighbors of a given vertex.
Let us now formulate our main result (recalled and proved below as Theorem 3).
Theorem 1
This equality is sharp for instance for a complete graph of n vertices where the left and the right hand side both equal to \(\frac{n2}{n1}\).
Without the triangle terms ♯(x,y) (which is the crucial term for our purposes), Theorem 1 is due to Lin–Yau [19, 21], and we take their proof as our starting point. Lin–Yau also obtain analogues of Bochner type inequalities and eigenvalue estimates as known from Riemannian geometry.
In Riemannian geometry, the Bochner formula encodes deep analytic properties of Ricci curvature. It is a key ingredient in proving many results, e.g. the spectral gap of the Laplace–Beltrami operator. A lower bound of the Ricci curvature implies a curvaturedimension inequality involving the Laplace–Beltrami operator through the Bochner formula. In an important work, Bakry and Émery [2, 3] generalize this inequality to generators of Markov semigroups, which works on measure spaces. Their inequality contains plentiful information and implies a lot of functional inequalities including spectral gap inequalities, Sobolev inequalities, and logarithmic Sobolev inequalities and many celebrated geometric theorems (see [1] and the references therein). Lin–Yau [21] study such inequalities on locally finite graphs.
In the present paper, we also want to find relations on locally finite graphs between Ollivier’s Ricci curvature and Bakry–Émery’s curvaturedimension inequalities, which represent the geometric and analytic aspects of graphs, respectively. Again, this is inspired by Riemannian geometry where one may attach a Brownian motion with a drift to a Riemannian metric [24]. We also mention that the definitions given by Sturm and Lott–Villani are also consistent with that of Bakry–Émery [22, 29, 30]. So exploring the relations on nonsmooth spaces may provide a good point of view to connect Ollivier’s definition to Sturm and Lott–Villani’s (in this respect, see also Ollivier–Villani [26]). In Sect. 4, we use the local clustering coefficient again to establish more precise curvaturedimension inequalities than those of Lin–Yau [21]. And with this in hand, we prove curvaturedimension inequalities under the condition that Ollivier’s Ricci curvature of the graph is positive.
Further analytical results following from curvaturedimension inequalities on finite graphs have been described in [19], and Lin–Lu–Yau [20] study a modified definition of Ollivier’s Ricci curvature on graphs. Recently, Paeng [27] studied upper bounds for the diameter and volume of finite simple graphs in terms of Ollivier’s Ricci curvature. For other works of synthetic Ricci curvatures on discrete spaces, see Dodziuk–Karp [14], Chung–Yau [9], Bonciocat–Sturm [7], and on cell complexes see Forman [17], Stone [28] etc.
We point out that, as in Riemannian geometry, both Ollivier’s Ricci curvature and Bakry–Émery’s curvaturedimension inequality can give lower bound estimates of the first eigenvalue λ _{1} for the Laplace operator (see Ollivier [24], Bakry [1]). Therefore our results in fact relate λ _{1} to the Watts–Strogatz local clustering coefficient, or the number of cycles with length 3. In [10], Diaconis and Stroock obtain several geometric bounds for eigenvalues of graphs, one of which is related to the number of odd length cycles. For more geometric quantities and methods concerning eigenvalue estimates in the study of Markov chains, see [11, 12, 13] and the references therein. We further explore the interaction between Ollivier’s Ricci curvature and eigenvalues estimates in joint work with Frank Bauer, see [6].
In this paper, G=(V,E) will denote an undirected connected simple graph without loops, where V is the set of vertices and E is the set of edges. V could be an infinite set. But we require that G is locally finite, i.e., for every x∈V, the number of edges connected to x is finite. For simplicity and in order to see more geometry, we mainly work on unweighted graphs. But we will also derive similar results on weighted graphs. In that case, we denote by w _{ xy } the weight associated to x,y∈V, where x∼y (we may simply put w _{ xy }=0 if x and y are not neighbors, to simplify the notation). The unweighted case corresponds to w _{ xy }=1 whenever x∼y. The degree of x∈V is d _{ x }=∑_{ y,y∼x } w _{ xy }.
2 Ollivier’s Ricci Curvature and Bakry–Émery’s Calculus
In this section, we present some basic facts about Ollivier’s Ricci curvature and Bakry–Emery’s Γ _{2} calculus, in particular on graphs.
2.1 Ollivier’s Ricci Curvature
Ollivier’s Ricci curvature works on a general metric space (X,d), on which we attach to each point x∈X a probability measure m _{ x }(⋅). We denote this structure by (X,d,m).
Definition 1
(Ollivier)
Here, W _{1}(m _{ x },m _{ y }) is the optimal transportation distance between the two probability measures m _{ x } and m _{ y }, defined as follows (cf. Villani [31, 32], Evans [16]).
Definition 2
Remark 1
A very important property of transportation distance is the Kantorovich duality (see, e.g. Theorem 1.14 in Villani [31]). We state it here in our particular graph setting.
Proposition 1
(Kantorovich Duality)
From this property, a good choice of a 1Lipschitz function f will yield a lower bound for W _{1} and therefore an upper bound for κ.
Remark 2

κ(x,y)≤1.

Rewriting (2.2) gives W _{1}(m _{ x },m _{ y })=d(x,y)(1−κ(x,y)), which is analogous to the expansion in the Riemannian case.
 A lower bound κ(x,y)≥k for any x,y∈X implieswhich can be seen as some kind of Lipschitz continuity of measures.$$ W_1(m_x, m_y)\leq(1k)d(x,y), $$(2.4)
2.2 Bakry–Émery’s CurvatureDimension Inequality
2.2.1 Laplace Operator
We will study the following operator which is an analogue of the Laplace–Beltrami operator in Riemannian geometry.
Definition 3
For our choice of {m _{ x }(⋅)}, this is the graph Laplacian studied by many authors, see e.g. [4, 5, 8, 14, 21].
2.2.2 Bochner Formula and CurvatureDimension Inequality
Definition 4
3 Ollivier’s Ricci Curvature and Triangles
In this section, we mainly prove lower bounds for Ollivier’s Ricci curvauture on locally finite graphs. In particular we shall explore the implication between lower bounds of the curvature and the number of triangles including neighboring vertices; the latter is encoded in the local clustering coefficient. We remark that we only need to bound κ(x,y) from below for neighboring x,y, since by the triangle inequality of W _{1}, this will also be a lower bound for κ(x,y) of any pair of x,y. (See Proposition 19 in Ollivier [24].)
3.1 Unweighted Graphs
In this subsection, we only consider unweighted graphs.
In Lin–Yau [21], they prove a lower bound of Ollivier’s Ricci curvature on locally finite graphs G. Here, for later purposes, we include the case where G may have vertices of degree 1 and get the following modified result.
Theorem 2
Remark 3
Notice that if d _{ x }=1, then we can calculate κ(x,y)=0 exactly. So, even though in this case \(2+\frac{2}{d_{x}}=0\), \(\kappa(x, y)\geq\frac{2}{d_{y}}\) does not hold.
For completeness, we state the proof of Theorem 2 here. It is essentially the one in Lin–Yau [21] with a small modification.
Proof of Theorem 2
Note that trees attain this lower bound. This coincides with the geometric intuition of curvature. Since trees have the fastest volume growth rate, it is plausible that they have the smallest curvature.
Proposition 2
Proof
In fact with Theorem 2 in hand, we only need to prove that \(1+2 (1\frac{1}{d_{x}}\frac{1}{d_{y}} )_{+}\) is also a lower bound of W _{1}. If one of x,y is a vertex of degree 1, say d _{ x }=1, it is obvious that W _{1}(m _{ x },m _{ y })=1. So we only need to deal with the case \(1\frac{1}{d_{x}}\frac{1}{d_{y}}\geq0\).
This completes the proof. □
Remark 4
Theorem 3
Remark 5
If ♯(x,y)=0, then this lower bound reduces to the one in Theorem 2.
Example 1

common neighbors of x,y: z∼x and z∼y:

x’s own neighbors: z∼x,z≁y,z≠y;

y’s own neighbors: z∼y,z≁x,z≠x.
Proof of Theorem 3
 1.
Move the mass of \(\frac{1}{d_{x}}\) from y to y’s own neighbors;
 2.
Move a mass of \(\frac{1}{d_{y}}\) from x’s own neighbors to x;
 3.
Fill gaps using the mass at x’s own neighbors. Filling the gaps at common neighbors costs 2 and the one at y’s own neighbors costs 3.
We will divide the discussion into three cases according to whether the first two steps can be realized or not.
Remark 6
Remark 7
Remark 8
From extending f to a 1Lipschitz function, we see that the paths of length 1 or 2 between neighbors of x and y have an important effect on the curvature. That is, in addition to triangles, quadrangles and pentagons are also related to Ollivier’s Ricci curvature. But polygons with more than five edges do not impact it.
Remark 9
If we see the graph G=(V,E) as a metric measure space (G,d,m), then the term ♯(x,y)/d _{ x }∨d _{ y } is exactly m _{ x }∧m _{ y }(G):=m _{ x }(G)−(m _{ x }−m _{ y })_{+}(G), i.e. the intersection measure of m _{ x } and m _{ y }. From a metric view, the vertices x _{1} that satisfy x _{1}∼x, x _{1}∼y constitute the intersection of the unit metric spheres S _{ x }(1) and S _{ y }(1).
From Theorem 3, we can force the curvature κ(x,y) to be positive by increasing the number ♯(x,y).
Theorem 4
Proof
We will denote D(x):=max_{ y,y∼x } d _{ y }. By the relation (3.7), we can get immediately
Corollary 1
Remark 10
3.2 Weighted Graphs
The preceding considerations readily extend to weighted graphs.
Theorem 5
Theorem 6
Remark 11
Proof
Theorem 7
4 CurvatureDimension Inequalities
In this section, we establish curvaturedimension inequalities on locally finite graphs. A very interesting one is the inequality under the condition κ≥k>0. Curvaturedimension inequalities on locally finite graphs are studied in Lin–Yau [21]. We first state a detailed version of their results. Let us denote \(D_{w}(x):=\max_{y, y\sim x}\frac{d_{y}}{w_{yx}}\). Notice that on an unweighted graph, this is the D(x) we used in Sect. 3.
Theorem 8
Remark 12
4.1 Unweighted Graphs
We again restrict ourselves to unweighted graphs.
We observe that the existence of triangles causes cancelations in calculating the term \(\operatorname{Hf}(x)\). This gives
Theorem 9
Remark 13
Notice that if there is a vertex y, y∼x, such that ♯(x,y)=0, this will reduce to (4.1).
Proof
Recalling Theorem 4 and the subsequent discussion, we get the following curvaturedimension inequalities on graphs with positive Ollivier–Ricci curvature.
Corollary 2
Corollary 3
Remark 14
Remark 15
We point out that the condition κ(x,y)≥k>0 implies that the diameter of the graph is bounded by \(\frac{2}{k}\) (see Proposition 23 in Ollivier [24]). So in this case the graph is a finite one.
To get a curvaturedimension inequality with a curvature term which behaves like κ, it seems that we should adjust the dimension parameter. In fact, we have
Proposition 3
Proof
Remark 16
Remark 17
In Erdös–Harary–Tutte [15], they define the dimension of a graph G as the minimum number n such that G can be embedded into a n dimensional Euclidean space with every edge of G having length 1. It is interesting that by their definition, the dimension of \(\mathcal{K}_{n}\) is also n−1 and the dimension of any tree is at most 2.
From the above observations, it seems natural to expect stronger relations between the lower bound of κ and the curvature term in the curvaturedimension inequality if one chooses proper dimension parameters.
4.2 Weighted Graphs
We have similar results on weighted graphs here, with similar proofs.
Theorem 10
Notes
Acknowledgements
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/20072013)/ERC grant agreement n^{∘}267087.
We thank Persi Diaconis for pointing out Ollivier’s notion of Ricci curvature to us.
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