Genealogical structures under interactive neutral reproduction: factorial moment duality via a Frankenstein process

TL;DR

This paper develops a genealogical framework linking a Wright--Fisher SDE and a counting process with interaction, introducing a novel 'Frankenstein process' to clarify complex genealogical structures.

Contribution

It constructs a new Moran model with interactive neutral reproduction and introduces the Frankenstein process to establish genealogical moment duality.

Findings

Established genealogical framework for moment duality

Constructed a Moran model with interactive neutral reproduction

Defined the Frankenstein process to simplify genealogical structures

Abstract

We establish a genealogical framework for an existing analytical moment duality between a Wright--Fisher type SDE and a counting process with interaction. To achieve this, we construct a finite-population Moran model featuring interactive neutral reproduction as a novel mechanism. In the corresponding events, an individual, regardless of its own type, can only reproduce if a randomly encountered partner is of the ``fit'' type. This Moran model has a relatively simple counting process as its factorial moment dual, whose genealogical meaning appears to be cryptic: after all, the line-counting process of the natural genealogical process of the model, namely the ancestral influence graph (AIG), exhibits a complex hierarchical structure not reflected in the factorial moment dual. Since moment duality is a property in expectation, we are allowed to systematically remove information from the AIG and merge different realizations of the ancestry. We call the result the \emph{Frankenstein process}. Based on this, we establish the factorial moment duality from a genealogical perspective. The moment duality in the diffusion limit follows in a natural way.