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To resolve whether multifurcations in sample Fréchet mean phylogenetic trees represent true population structure or non-Euclidean statistical artifacts ("stickiness" in Billera-Holmes-Vogtmann space), this work models the Fréchet mean process via an embedded multidimensional random walk. The authors prove that the exit time from a lower-dimensional stratum corresponds to the maximum last-passage time above zero of the random walk's coordinates, yielding a fully nonparametric estimator for whether sample trifurcations will bifurcate given more data. Applying the method to the evolutionary radiation of primates, glires, and tree shrews provides a rigorous diagnostic for distinguishing genuine hard polytomies from sample-starved soft polytomies.
Spurious branch collapses in non-Euclidean tree spaces are often geometric traps of Fréchet means—and can now be diagnosed via the exact last-passage times of an embedded random walk.
A well-known phenomenon in statistical analyses of populations of phylogenetic trees in the Billera-Holmes-Vogtmann space is that the topology of the Fr\'echet mean tree can contain multifurcations (i.e., internal nodes with more than two children), which raises the practical question of whether this reflects a population-level branching structure (hard polytomy) or merely sampling variability in the data (soft polytomy). This is an instance of the more general phenomenon of"stickiness"in non-Euclidean statistics, whereby the sample Fr\'echet mean in certain non-positively curved stratified spaces becomes permanently trapped in a lower-dimensional stratum. In this work, we identify a particular multidimensional random walk embedded within the Fr\'echet mean process, and we show that the time at which stickiness occurs is determined by the largest last-passage time above zero of the coordinates of this random walk. Using this representation, we develop a fully nonparametric procedure for estimating the probability that trifurcations in a sample Fr\'echet mean tree will bifurcate at some future time if more observations are collected. Lastly, we apply our methodology to a problem in phylogenetics where we consider whether an observed trifurcation in the species tree of primates, glires, and tree shrews is genuinely trifurcated at the population level.
BAIR