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Ancestral recombination graphs
Mathematical and visualization tools for working with ancestral recombination graphs
The fields of phylogenetics and population genetics share several important models including gene trees, species trees, ancestral recombination graphs (ARGs), and pedigrees. These models are all closely related and can be viewed as subgraphs of one another. Amongst them, the ARG is particularly central and if inferred efficiently can enable many applications such as inference of selection and demography. Here, I will review various helpful mathematical tools for working with ARGs, including what we call the threading algorithm, the branch graph, and the leaf trace visualization.
An empirical view of the population pedigree
Often, the summary statistics of population genetics are framed in the setting of Kingman's coalescent or related models. These statistics can be alternatively thought of as descriptive statistics of the realized population pedigree-with-recombination, in a way that has become much more useful in the era of whole-genome sequencing. For instance, pairwise number of nucleotide differences is proportional to "effective population size", which is sometimes more usefully thought of as an estimate of the average length of the path through the pedigree to the most recent common ancestor at a randomly chosen locus (with an explicit standard error). Another example is the pairwise distribution of long tracts of IBD, which provides an estimate of a functional of the entire distribution of such paths.