This is a gentle introduction to the context and results of my article ricci curvature of markov chains on metric spaces. Discrete curvature and abelian groups canadian journal of. Ricci curvature bounds in the sense of lott, sturm, and villani is still under active development 2, 3. The goal is to present a notion of ricci curvature valid on arbitrary metric spaces, such as graphs, and to generalize a series of classical. Pdf ricci curvature of the internet topology emil saucan. Yuval peresy oded schrammz scott sheeldx october 19, 2004 abstract a metric space x has markov type 2, if for any reversible. Ollivier proposed in 2009 a curavture notion of markov chains on metric spaces, based on optimal transport of probability measures associated to a random walk. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the ornsteinuhlenbeck process. Entropic ricci curvature bounds for discrete interacting systems fathi, max and maas, jan, the annals of applied probability, 2016. This notion is inspired by the seminal work of lott, sturm and villani, who developed a synthetic notion of ricci curvature for geodesic spaces based on convexity of the entropy along 2wasserstein geodesics. Optimal transport and ricci curvature for metricmeasure spaces. We define the ricci curvature of markov chains on metric spaces as a local contraction coefficient of the random walk acting on the space of probability measures equipped with a wasserstein transportation distance.
In 2009, ollivier 20 gave a notion of coarse ricci curvature of markov chains valid on arbitrary metric spaces, such as graphs. Curvature and transport inequalities for markov chains in discrete. It begins with a description of classical ricci curvature in riemannian geometry, as well as a reminder for discrete markov chains. Ricci curvature of finite markov chains via convexity of. Optimal transport and ricci curvature for metric measure spaces john lott abstract. Three generalized notions of curvature there are three candidates for generalized version of the ricci curvature. The state space is the set of possible values for the observations. Quantifying the systemic risk and fragility of financial systems is of vital importance in analyzing market efficiency, deciding on portfolio allocation, and containing financial contagions. Entropic ricci curvature bounds for discrete interacting. It has been shown in 9 that a ricci curvature lower bound has significant consequences for the associated markov chain. On the equivalence of the entropic curvaturedimension condition and bochners inequality on metric measure spaces.
As we shall see, we can associate a notion of curvature to a markov chain. For a riemannian manifold this gives back, after scaling, the value of ricci curvature of a tangent vector. Theorem 3 let xbe an geodesic metric space and p be the transition kernel of a markov chain on x. Oliviers ricci curvature and applications on cell networks. A markov chain is then said to have ricci curvature bounded from below by some constant k if the relative entropy with respect to the invariant probability measure of the chain is k convex along geodesies for the metric w. Finally, ollivier 16 introduced a notion of coarse ricci curvature that extends to any markov chains on metric spaces. A simple differential geometry for networks and its. The goal of this paper is to demonstrate that ricci curvature may serve as a quantitative indicator of the systemic risk in financial networks and the fragility of financial markets. Introduction the purpose of this note is to give a liouville type theorem for harmonic map to metric space having a convex property in terms of discrete time markov chains.
The notion of entropic ricci curvature on discrete spaces i. A survey of ricci curvature for metric spaces and markov chains ollivier, yann, 2010. This text is a presentation of the general context and results of oll07 and oll09, with comments on related work. We prove wasserstein contraction and a lichnerowiczlike spectral gap bound for reversible markov processes with positive coarse ricci curvature. We survey work of lottvillani and sturm on lower ricci curvature bounds for metric measure spaces. We study a natural discrete bochnertype inequality on graphs, and explore its merit as a notion of curvature in discrete spaces. The necessary background concentration of measure, curvature in riemannian geometry, convergence of markov chains is covered in the first section. Notions of ricci curvature have been particularly successful. Exact and asymptotic results on coarse ricci curvature of.
Ricci curvature for metricmeasure spaces via optimal transport. We study a new notion of ricci curvature that applies to markov chains on discrete spaces. Optimal transport and ricci curvature for metricmeasure. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Curvature, in the broad sense, is a measure by which a geometrical object deviates from being flat. We note moreover, that that the riemannian metric tensor associate to w can be. The time evolution of adaptive algorithms depends on past samples, and thus these algorithms are nonmarkovian.
Such a space is said to have ricci curvature bounded below by, provided the relative. Quantum ricci curvature has been introduced recently as a new, geometric observable characterizing the curvature properties of metric spaces, without the need for a smooth structure. In some moral sense, the boundary points are metric spaces with nonnegative ricci curvature. Such a space is said to have ricci curvature bounded below by, provided the relative entropy is convex along geodesics in the wasserstein space of probability measures. Ollivierricci curvature as a natural metric to discover hierarchies in graphs. In order to apply to the discrete setting, the role of the wasserstein metric is taken over by. Osaka city university ricci curvature of markov chains on polish spaces revisited. The proposed community detection algorithm utilizes the coarse ricci curvature, referred to as ollivierricci curvature, in its version designed for complex networks. The coarse ricci curvature of the markov chain is said to be bounded from below by. Conference analysis and probability this text is a presentation of the general context and results of oll07 and the preprint olla, with comments on related work. Ricci curvature of markov chains via convexity of the entropy jan maas university of bonn 6icsa bedlewo 10 september 2012 joint work with matthias erbar bonn. Curvature and transport inequalities for markov chains in.
We study functional inequalities for markov chains on discrete spaces with en. This definition naturally extends to any markov chain on a metric space. In order to apply to the discrete setting, the role of the wasserstein metric is taken over by a different metric, having the property that continuous. On spectral bounds for symmetric markov chains with coarse ricci. Ricci bounds for interacting markov chains been taken by sturm 22 and lott and villani 14 and applies to metric measure spaces. Oct 14, 2019 this came about because it was realized that notions that originated in differential and riemannian geometry possess a wider geometric meaning that extends to metric spaces, and therefore, in particular, to graphs. Ricci curvature of finite markov chains via convexity of the entropy matthias erbar and jan maas abstract. Other generalizations of ricci curvature start with a metric measure space 31,46. On spectral bounds for symmetric markov chains with coarse. Framework throughout this note, y, d denotes a complete separable. The boundary points are compact metric spaces x,d with dimh x. In the special setting of graphs, this concept provides a curvature on the edges and depends on an idleness parameter of the random walk. The goal is to present a notion of ricci curvature valid on arbitrary metric spaces, such as graphs, and to generalize a series of classical theorems in positive ricci curvature, such as spectral gap estimates, concentration of measure or logsobolev inequalities.
Ollivierricci curvaturebased method to community detection. Our main results are that when curvature is nonnegative, but not necessarily positive, the spectral gap, the cheeger isoperimetric constant and the modified logarithmic sobolev constant of the chain can be bounded from below by a constant that only depends on the. We present a notion of ricci curvature that applies to markov chains on discrete spaces. In general, the curvature is a way of measuring the degree to which the geometry determined by a given riemannian metric might di er from that of ordinary euclidean space. Coarse ricci curvature for continuoustime markov processes. Transportation and concentration inequalities for bifurcating markov chains bitseki penda, s. Combinatorial ricci curvature on cellcomplex and gaussbonnnet theorem watanabe, kazuyoshi, tohoku mathematical journal, 2019. Ricci curvature on general spaces without riemannian structures has been recently studied, in the work of ollivier 19, 20 on markov chains, and bakry and emery 38, lott, villani 21, bonciocat and sturm 39, 40 on general metric.
In particular, if we let x, d be metric space equipped with a family of probability measures. Examples of positively curved spaces for this definition include the. In this paper, we show that the ricci curvature lower bound in olliviers wasserstein metric sense of a continuous time jumping markov process on a graph can be characterized by some optimal. Transportentropy inequalities and curvature in discrete. We define the ricci curvature of markov chains on metric spaces as a local contraction coefficient of the random walk acting on the space of probability. Aug 22, 2012 we study a new notion of ricci curvature that applies to markov chains on discrete spaces. Since the ricci curvature involves a privileged direction. Ricci curvature of markov chains on metric spaces sciencedirect. Itisreversible if moreover, the detailed balance condi tion d. Poissontype deviation inequalities for curved continuous. In the discrete setting the role of the 2wasserstein metric is taken over by a. Ricci curvature of markov chains on metric spaces core. For this we use a local contraction coefficient of the random walk acting on the space of probability measures equipped with a transportation distance. At a high level, financial systems may be represented as weighted graphs that characterize the complex web of interacting agents and information flow for example, debt, stock returns, and shareholder.
Besides coordinate invariance, its key features are scalability, computability and robustness. Recently, ollivier 9 introduced the ricci curvature of markov chains on polish spaces, which was characterized by the contractivity of transition probabilities under the l. We modify the definition of ricci curvature of ollivier of markov chains on graphs to study the properties of the ricci curvature of general graphs, cartesian product of graphs, random graphs, and. Ricci curvature of markov chains via convexity of the entropy.
This was used to generalize a series of classical theorems in positive ricci curvature, such as spectral gap estimates, concentration of measure or logsobolev inequalities 15, 17. We define a notion of ricci curvature in metric spaces equipped with a measure or a random walk. In analogy with curvature notions in riemannian geometry, we interpret this ricci curvature as a con. Adaptive markov chains are an important class of monte carlo methods for sampling from probability distributions. A crucial feature of the theory is its robustness, i.
Each interior point is a riemannian manifold m,g with dimm n, diamm. Ricci curvature of metric spaces university of chicago. Some recent results for ollivier ricci curvature on graphs. The curvature is a property of both the metric space and the markov kernel. Ollivier, ricci curvature of markov chains on metric spaces, preprint, arxiv.
This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by lott, sturm, and villani for geodesic measure spaces. Here we will prove the following concentration result for the equilibrium measure of markov chains. However, it is not just a space that can have positive or otherwise curvature. In the course of obtaining a good logsobolev inequality, they found the. 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. Markov chain monte carlo, concentration of measure, ricci curvature. We define a notion of ricci curvature in metric spaces equipped with a. This definition provides a synthetic notion of ricci curvature of markov chains on metric spaces and its geometric validation comes in the form of analogues of the classical theorems of. Mixing and concentration by ricci curvature request pdf. Those notations of ricci curvature work on so called length spaces.
Moderate deviations for empirical measures of markov chains. A concentration theorem for the equilibrium measure of. Ricci curvature of finite markov chains via convexity of the. Discrete ricci curvature via convexity of the entropy. We demonstrate that these properties continue to hold in the context of nonperturbative quantum gravity, by evaluating. An intriguing question is whether one can extend notions of smooth riemannian geometry to general metric spaces. Markov chains in smooth banach spaces and gromov hyperbolic metric spaces assaf naor. For a riemannian manifold this gives back the value of ricci curvature. Constant curvature metrics for markov chains nasaads. Ricci curvature for metricmeasure spaces via optimal. A survey of ricci curvature for metric spaces and markov.
Given a metric space carrying a probability distribution, this is an interesting problem to construct a invariant markov chain with the highest possible curvature. For brownian motion on a riemannian manifold this gives back the value of ricci curvature of a tangent vector. A survey of ricci curvature for metric spaces and markov chains. Volume 57 of advanced studies in pure mathematics the mathematical society of japan, tokyo, 2010, pp. Gromovhausdorff space each point is a compact metric space. Olliviers ricci curvature, local clustering and curvature.
There are some recent works on giving a good notion for a metric measure space to have a lower ricci curvature bound, see 21, 18 and 19. In riemannian geometry, ricci curvature is the relevant notion in a series of positivecurvature theorems see section 1. Although the various brownian curvatures on a smooth riemannian manifold are essentially equivalent and. Ricci curvature bounds for weakly interacting markov chains. Ollivier, a survey of ricci curvature for metric spaces and markov chains, in probabilistic approach to geometry. Ricci curvature of markov chains on metric spaces citeseerx. Thus, for the example above the state space consists of two states. Ricci curvature lower bounds for geodesic measure spaces.
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