# Ricci Curvature Of Markov Chains On Metric Spaces Homework

## Title: Ricci curvature of Markov chains on metric spaces

Authors:Yann Ollivier

(Submitted on 30 Jan 2007 (v1), last revised 30 Jul 2007 (this version, v4))

Abstract: 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. For Brownian motion on a Riemannian manifold this gives back 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--\'Emery Ricci curvature for Brownian motion with a drift on a Riemannian manifold.

Positive Ricci curvature is easily shown to imply a spectral gap, a L\'evy--Gromov-like Gaussian concentration theorem and a kind of modified logarithmic Sobolev inequality. These bounds are sharp in several interesting examples.

## Submission history

From: Yann Ollivier [view email]**[v1]**Tue, 30 Jan 2007 16:41:12 GMT (22kb)

**[v2]**Wed, 28 Feb 2007 20:56:04 GMT (28kb)

**[v3]**Fri, 1 Jun 2007 13:50:39 GMT (48kb)

**[v4]**Mon, 30 Jul 2007 17:57:20 GMT (58kb)

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Statistical phylogenetic inference methods use tree rearrangement operations such as subtree–prune–regraft (SPR) to perform Markov chain Monte Carlo (MCMC) across tree topologies. The structure of the graph induced by tree rearrangement operations is an important determinant of the mixing properties of MCMC, motivating the study of the underlying *SPR graph* in greater detail.

In this paper, we investigate the SPR graph of rooted trees (rSPR graph) in a new way: by calculating the Ricci–Ollivier curvature with respect to uniform and Metropolis–Hastings random walks. This value quantifies the degree to which a pair of random walkers from specified points move towards each other; negative curvature means that they move away from one another on average, while positive curvature means that they move towards each other. In order to calculate this curvature, we develop fast new algorithms for rSPR graph computation. We then develop formulas characterizing how the number of rSPR neighbors of a tree changes after an rSPR operation is applied to that tree. These give bounds on the curvature, as well as a flatness-in-the-limit theorem indicating that paths of small topology changes are easy to traverse. However, we find that large topology changes (i.e. moving a large subtree) give pairs of trees with negative curvature. We show using simulation that mean access time distributions depend on distance, degree, and curvature, demonstrating the relevance of these results to stochastic tree search.

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