Time-Dependent Estimates of Phylogenetic Diversification.
Time-dependent estimates of phylogenetic diversification were measured by extracting the branches descending from each internal node (above the root) of each phylogeny and calculating the peak height (η) of the spectral density profile of the graph Laplacian of each subtree, which is a measure of the density of branching events (36, 37). The code to perform the analysis is available for download at https://www.hivresearch.org/publication-supplements.

Simulation analyses.
Phylogenies were simulated using a time-forward branching process under constant birth rates (b(t)=b) and time-dependent birth rates (b(t)=beαt) for b = 0.01, 0.03, 0.05, 0.07, and 0.09 and α = ±0.01, ±0.11, ±0.21, ±0.31, and ±0.41, for 20, 220, 420, 620, and 820 tips, and for 1, 11, 21, 31, and 41 time units. Simulated phylogenies were downsampled at 0%, 10%, 30%, 50%, and 70%. For each scenario, 100 phylogenies were simulated. Time-dependent diversification (i.e., η across subtrees) was calculated for each phylogeny simulated under each scenario. Simulations were conducted using the R packages RPANDA (R Phylogenetic ANalyses of DiversificAtion) (61) and ape (53).

Comparisons to SARS-CoV-2 phylogeny.
Phylogenies downsampled at 10% from the full (18,514 tips) SARS-CoV-2 genome phylogeny (following the subsampling strategy described above) were used to calculate the phylogenetic η for each subtree (above the root) for each downsampled phylogeny. Neutral phylogenies were simulated under stochastic branching by randomly sampling from the distribution of branch lengths from one downsampled SARS-CoV-2 phylogeny. This was iterated across all downsampled SARS-CoV-2 phylogenies. Positive time-dependent phylogenies were simulated using a time-dependent process (b(t)=0.01eαt) for α = 0.001, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 1, with branch lengths restricted to the distribution of branch lengths from one downsampled SARS-CoV-2 phylogeny. This was iterated across all downsampled SARS-CoV-2 phylogenies for each α. Neutral and positive time-dependent phylogenies were simulated with a 10% sampling fraction. Polytomies were randomly resolved. Simulations were conducted using the R packages RPANDA (61) and ape (53).