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

摘要

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~2n)-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 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.