As a result of recombination, adjacent nucleotides can have different paths of genetic inheritance and therefore the genealogical trees for a sample of DNA sequences vary along the genome. The structure capturing the details of these intricately interwoven paths of inheritance is referred to as an ancestral recombination graph (ARG). New developments have made it possible to infer ARGs at scale, enabling many new applications in population and statistical genetics. This rapid progress, however, has led to a substantial gap opening between theory and practice. Standard mathematical formalisms, based on exhaustively detailing the "events" that occur in the history of a sample, are insufficient to describe the outputs of current methods. Moreover, we argue that the underlying assumption that all events can be known and precisely estimated is fundamentally unsuited to the realities of modern, population-scale datasets. We propose an alternative mathematical formulation that encompasses the outputs of recent methods and can capture the full richness of modern large-scale datasets. By defining this ARG encoding in terms of specific genomes and their intervals of genetic inheritance, we avoid the need to exhaustively list (and estimate) all events. The effects of multiple events can be aggregated in different ways, providing a natural way to express many forms of approximate and partial knowledge about the recombinant ancestry of a sample.

A general and efficient representation of ancestral recombination graphs