Ricci-Ollivier Curvature of the Rooted Phylogenetic Subtree-Prune-Regraft Graph

Abstract

Statistical phylogenetic inference methods use tree rearrangement operations such as subtree-prune-regraft (SPR) to perform Markov chain Monte Carlo (MCMC) across tree topologies. These methods are known to mix quickly when sampling from the simple uniform distribution of trees but may become stuck in the local optima of multi-modal posterior distributions for real data induced by non-uniform likelihoods. The structure of the graph induced by tree rearrangement operations is an important determinant of the mixing properties of MCMC, motivating study of the underlying rSPR graph in greater detail.In this paper, we investigate the rSPR graph in a new way: by calculating Ricci-Ollivier curvature with respect to uniform and Metropolis-Hastings random walks. We confirm using simulation that mean access time distributions depend on distance, degree, and curvature, showing the relevance of these curvature results to stochastic tree search. These calculations require fast new algorithms for constructing and sampling these graphs, reducing the time required to compute an rSPR graph from O(m 2 n)-time to O(mn 3 ), where m is the (often large) number of trees in the graph and n their number of leaves, and reducing the time required to select an SPR neighbor of a tree uniformly at random to O(n) time. We then develop a closed form solution to characterize how the number of SPR neighbors of a tree changes after an SPR operation is applied to that tree. This gives 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) gives pairs of trees with negative curvature. Although these pairs of trees with negative curvature do not impede mixing in this simple well-connected space, they may manifest as bottlenecks in the much smaller * This work was funded by National Science Foundation award 1223057. Chris Whidden is a Simons Foundation Fellow of the Life Sciences Research Foundation.† Program in Computational Biology, Fred Hutchinson Cancer Research Center, Seattle, WA, USA 98109. {cwhidden,matsen}@fredhutch.org credible sets induced by phylogenetic posteriors with a likelihood function. This work extends our knowledge of the rSPR graph, in particular properties that are relevant for investigation of sampling the rSPR graph.

Tanglegrams: A Reduction Tool for Mathematical Phylogenetics

“…We observe that many problems in phylogenetic combinatorics "factor" through a problem on tanglegrams. As such, we believe tanglegrams to be a worthwhile object of study in phylogenetic combinatorics, and note that they have already been crucial in an analysis of the geometry of the subtree-prune-regraft graph [11].…”

“…The same is true for more complex random walks such as Markov chain Monte Carlo using a label-invariant likelihood, as would be used for sampling from a prior distribution on trees [9]. Graph characteristics such as Ricci-Ollivier curvature [10] under simple random walks or MCMC with a label-invariant likelihood are expressed in terms of relative tree labelings [11]. Analogous considerations hold for the problem of species delimitation, which can naturally be phrased in terms of inference of a partition of relatively labeled objects: neither distances between partitions [12] nor the graphs underlying MCMC over these partitions [13] actually refer to labels themselves.…”

Lagged couplings diagnose Markov chain Monte Carlo phylogenetic inference

“…They find that topologies with high posterior support are frequently isolated from each other by multiple SPR moves; that is, phylogenetic posteriors are naturally multimodal when distance is measured by the number of MCMC steps to transition between states. Whidden and Matsen IV (2017) further analyse properties of the SPR graph. Chains which move more frequently produce lower asymptotic variances (Peskun, 1973) so some form of adaptive MCMC algorithm (Haario et al, 2001) is a component of many standard phylogenetics software packages.…”

Modern-Day SIV Viral Diversity Generated by Extensive Recombination and Cross-Species Transmission

“…HIV-2 has been acquired through at least 8 independent spillover events from sooty mangabeys [5]. Notably, 6 of these transmissions have resulted in only a single observed infection (spillovers) [7,8,36]; only 2 of these events have established sustained transmission chains and successfully switched hosts to become endemic human pathogens [37][38][39][40]. Given this pattern, it is conceivable that there have been many isolated introductions of lentiviruses into humans over the past 200,000 years [14,41].…”