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

Abstract

Statistical phylogenetic inference methods use tree rearrangement operations to perform either hill-climbing local search or Markov chain Monte Carlo across tree topologies. The canonical class of such moves are the subtree-prune-regraft (SPR) moves that remove a subtree and reattach it somewhere else via the cut edge of the subtree. Phylogenetic trees and such moves naturally form the vertices and edges of a graph, such that tree search algorithms perform a (potentially stochastic) traversal of this SPR graph. Despite the centrality of such graphs in phylogenetic inference, rather little is known about their large-scale properties. In this paper we learn about the rooted-tree version of the graph, known as the rSPR graph, by calculating the Ricci-Ollivier curvature for pairs of vertices in the rSPR graph with respect to two simple random walks on the rSPR graph. By proving theorems and direct calculation with novel algorithms, we find a remarkable diversity of different curvatures on the rSPR graph for pairs of vertices separated by the same distance. We confirm using simulation that degree and curvature have the expected impact on mean access time distributions, demonstrating relevance of these curvature results to stochastic tree search. This indicates significant structure of the rSPR graph beyond that which was previously understood in terms of pairwise distances and vertex degrees; a greater understanding of curvature could ultimately lead to improved strategies for tree search.