Coarse Ricci Curvature Research Articles

Asymptotic coarse Ricci curvature of inhomogeneous random graph

In this paper, we study the coarse Ricci curvature of inhomogeneous random graph on vertex set [n]≔{1,2,…,n}. In this graph, each pair of vertices (i,j) forms an edge independently with probability λnαgin,jn for some function g(x,y)∈(0,1], constants λ>0 and α∈[0,1]. We derive the asymptotic coarse Ricci curvature of this random graph. Phase transition phenomenon exists as α varies from zero to one.

Transport distance between Grover walks on graphs and coarse Ricci curvature

Transport distance between Grover walks on graphs and coarse Ricci curvature

Coarse Ricci curvature of quantum channels

Following Ollivier's work [61], we introduce the coarse Ricci curvature of a quantum channel as the contraction coefficient of non-commutative metrics on the state space. These metrics are defined as a non-commutative transportation cost in the spirit of [42,41], which gives a unified approach to different quantum Wasserstein distances in the literature. We prove that the coarse Ricci curvature lower bound and its dual gradient estimate, under suitable assumptions, imply the Poincaré inequality (spectral gap) as well as transportation cost inequalities. Using intertwining relations, we obtain positive coarse Ricci curvature bounds of Gibbs samplers, Bosonic beam-splitters as well as Pauli channels on n-qubits.

Cheng's maximal diameter theorem for hypergraphs

We prove that Cheng's maximal diameter theorem for hypergraphs with positive coarse Ricci curvature. Coarse Ricci curvature on hypergraphs is defined by using the resolvent of the nonlinear Laplacian. As a byproduct of the main theorem, the first non-zero eigenvalue on a hypergraph coincides with the lower bound of the curvature if the hypergraph has the maximal diameter.

Quantum Concentration Inequalities

We establish Transportation Cost Inequalities (TCIs) with respect to the quantum Wasserstein distance by introducing quantum extensions of well-known classical methods: First, we generalize the Dobrushin uniqueness condition to prove that Gibbs states of 1D commuting Hamiltonians satisfy a TCI at any positive temperature and provide conditions under which this first result can be extended to non-commuting Hamiltonians. Next, using a non-commutative version of Ollivier’s coarse Ricci curvature, we prove that high temperature Gibbs states of commuting Hamiltonians on arbitrary hypergraphs H=(V,E) satisfy a TCI with constant scaling as O(|V|). Third, we argue that the temperature range for which the TCI holds can be enlarged by relating it to recently established modified logarithmic Sobolev inequalities. Fourth, we prove that the inequality still holds for fixed points of arbitrary reversible local quantum Markov semigroups on regular lattices, albeit with slightly worsened constants, under a seemingly weaker condition of local indistinguishability of the fixed points. Finally, we use our framework to prove Gaussian concentration bounds for the distribution of eigenvalues of quasi-local observables and argue the usefulness of the TCI in proving the equivalence of the canonical and microcanonical ensembles and an exponential improvement over the weak Eigenstate Thermalization Hypothesis.

A New Transport Distance and Its Associated Ricci Curvature of Hypergraphs

Abstract The coarse Ricci curvature of graphs introduced by Ollivier as well as its modification by Lin–Lu– Yau have been studied from various aspects. In this paper, we propose a new transport distance appropriate for hypergraphs and study a generalization of Lin–Lu–Yau type curvature of hypergraphs. As an application, we derive a Bonnet–Myers type estimate for hypergraphs under a lower Ricci curvature bound associated with our transport distance. We remark that our transport distance is new even for graphs and worthy of further study.

Condensed Ricci curvature of complete and strongly regular graphs

We study a modified notion of Ollivier's coarse Ricci curvature on graphs introduced by Lin, Lu, and Yau in [11]. We establish a rigidity theorem for complete graphs that shows a connected finite simple graph is complete if and only if the Ricci curvature is strictly greater than one. We then derive explicit Ricci curvature formulas for strongly regular graphs in terms of the graph parameters and the size of a maximal matching in the core neighborhood. As a consequence we are able to derive exact Ricci curvature formulas for strongly regular graphs of girth 4 and 5 using elementary methods. An example is provided that shows there is no exact formula for the Ricci curvature for strongly regular graphs of girth $3$ that is purely in terms of graph parameters.

An estimate of the first non-zero eigenvalue of the Laplacian by the Ricci curvature on edges of graphs

We define the distance between edges of graphs and study the coarse Ricci curvature on edges. We consider the Laplacian on edges based on the Jost-Horak's definition of the Laplacian on simplicial complexes. As one of our main results, we obtain an estimate of the first non-zero eigenvalue of the Laplacian by the Ricci curvature for a regular graph.

Lower Bound for the Coarse Ricci Curvature of Continuous-Time Pure-Jump Processes

We obtain a lower bound for the coarse Ricci curvature of continuous-time pure-jump Markov processes, with an emphasis on interacting particle systems. Applications to several models are provided, with a detailed study of the herd behavior of a simple model of interacting agents.

Curvature and transport inequalities for Markov chains in discrete spaces

We study various transport-information inequalities under three different notions of Ricci curvature in the discrete setting: the curvature-dimension condition of Bakry and Emery (In Seminaire de Probabilites, XIX, 1983/84 (1985) 177–206 Springer), the exponential curvature-dimension condition of Bauer et al. (Li-Yau Inequality on Graphs (2013)) and the coarse Ricci curvature of Ollivier (J. Funct. Anal. 256 (2009) 810–864). We prove that under a curvature-dimension condition or coarse Ricci curvature condition, an $L_{1}$ transport-information inequality holds; while under an exponential curvature-dimension condition, some weak-transport information inequalities hold. As an application, we establish a Bonnet–Myers theorem under the curvature-dimension condition $\operatorname{CD}(\kappa,\infty)$ of Bakry and Emery (In Seminaire de Probabilites, XIX, 1983/84 (1985) 177–206 Springer).

Coarse Ricci Curvature as a Function on $$\varvec{M\times M}$$

We use the framework used by Bakry and Emery in their work on logarithmic Sobolev inequalities to define a notion of coarse Ricci curvature on smooth metric measure spaces alternative to the notion proposed by Y. Ollivier. This function can be used to recover the Ricci tensor on smooth Riemannian manifolds by the formula $$\begin{aligned} \mathrm {Ric}(\gamma ^{\prime }\left( 0\right) ,\gamma ^{\prime }\left( 0\right) )=\frac{1}{2}\frac{{d}^{2}}{{d}s^{2}}\mathrm {Ric}_{\triangle _{g} }(x,\gamma \left( s\right) ) \end{aligned}$$ for any curve $$\gamma (s)$$ .

The Ricci curvature of a weighted tree

An expression for the coarse Ricci curvature of a weighted tree with a random walk on the vertex set is obtained. As a corollary, it is shown that the structure of a binary tree can be reconstructed up to isomorphism from the matrix of pairwise Ricci curvatures of its vertices.

The estimate of the Ricci curvature of a weighted tree

For weighed trees with random walk on the vertex set an estimate of the coarse Ricci curvature is obtained.

Mixing and concentration by Ricci curvature

We generalise the coarse Ricci curvature method of Ollivier by considering the coarse Ricci curvature of multiple steps in the Markov chain. This implies new spectral bounds and concentration inequalities. We also extend this approach to the bounds for MCMC empirical averages obtained by Joulin and Ollivier. We prove a recursive lower bound on the coarse Ricci curvature of multiple steps in the Markov chain, making our method broadly applicable. Applications include the split–merge random walk on partitions, Glauber dynamics with random scan and systemic scan for statistical physical spin models, and random walk on a binary cube with a forbidden region.

Ricci curvature of a weighted tree

A formula for the coarse Ricci curvature of a weighed tree with a random walk on vertex set is obtained. A criterion of restoration of a binary tree topology from its Ricci curvature matrix is proved as a corollary.

Exact and asymptotic results on coarse Ricci curvature of graphs

Ricci curvature was proposed by Ollivier in a general framework of metric measure spaces, and it has been studied extensively in the context of graphs in recent years. In this paper we obtain the exact formulas for Ollivier’s Ricci-curvature for bipartite graphs and for the graphs with girth at least 5. These are the first formulas for Ricci-curvature that hold for a wide class of graphs, and extend earlier results where the Ricci-curvature for graphs with girth 6 was obtained. We also prove a general lower bound on the Ricci-curvature in terms of the size of the maximum matching in an appropriate subgraph. As a consequence, we characterize the Ricci-flat graphs of girth 5. Moreover, using our general lower bound and the Birkhoff–von Neumann theorem, we give the first necessary and sufficient condition for the structure of Ricci-flat regular graphs of girth 4. Finally, we obtain the asymptotic Ricci-curvature of random bipartite graphs G(n,n,p) and random graphs G(n,p), in various regimes of p.

Ricci Curvature on Polyhedral Surfaces via Optimal Transportation

The problem of correctly defining geometric objects, such as the curvature, is a hard one in discrete geometry. In 2009, Ollivier 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 and Yau and Jost and Liu 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.

Lower bound of coarse Ricci curvature on metric measure spaces and eigenvalues of Laplacian

In this paper,we investigate the coarse Ricci curvature on metric spaces with random walks. There exists no canonical random walk on metric space with a reference measure. However, we prove that a Bishop–Gromov inequality gives a lower bound of coarse Ricci curvature with respect to a random walk called an $$r$$ -step random walk. The lower bound does not coincide with the constant corresponding to curvature in Bishop–Gromov inequality. As a corollary, we obtain a lower bound of coarse Ricci curvature with respect to an $$r$$ -step random walk for a metric measure space satisfying the curvature-dimension condition. Moreover we give an important example, Heisenberg group, which does not satisfy the curvature-dimension condition for any constant but has a lower bound of coarse Ricci curvature. We also have an estimate of the eigenvalues of the Laplacian by a lower bound of coarse Ricci curvature.

A Curved Brunn--Minkowski Inequality on the Discrete Hypercube, Or: What Is the Ricci Curvature of the Discrete Hypercube?

We compare two approaches to Ricci curvature on nonsmooth spaces in the case of the discrete hypercube $\{0,1\}^N$. While the coarse Ricci curvature of the first author readily yields a positive value for curvature, the displacement convexity property of Lott, Sturm, and Villani could not be fully implemented. Yet along the way we get new results of a combinatorial and probabilistic nature, including a curved Brunn--Minkowski inequality on the discrete hypercube.

Ricci curvature of Markov chains on metric spaces

We define the coarse Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are. This definition naturally extends to any Markov chain on a metric space. For a Riemannian manifold this gives back, after scaling, the value of Ricci curvature of a tangent vector. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the Ornstein–Uhlenbeck process. Moreover this generalization is consistent with the Bakry–Émery Ricci curvature for Brownian motion with a drift on a Riemannian manifold. Positive Ricci curvature is shown to imply a spectral gap, a Lévy–Gromov–like Gaussian concentration theorem and a kind of modified logarithmic Sobolev inequality. The bounds obtained are sharp in a variety of examples.