Communication Dans Un Congrès Année : 2020

A Mathematical Model for the Ancestor Paradox

Résumé

Each individual has two parents, hence four grandparents, eight great-grandparents, and so on. It thus seems natural to expect that by going back n generations into the past, the individual has 2^n ancestors at this generation. This number grows very rapidly, however, and for a sufficiently large n might even exceed the number of individuals alive at that time. For instance, 2^30 (which is more than one billion) exceeds the worldwide population 30 generations ago (a few hundred million). The flaw in this naive reasoning is that one ignores the fact that some ancestors might be counted several times. When people marry relatives (with whom they share ancestors, by definition), their progeny will have fewer ancestors than this original reasoning suggests. For instance, Ferdinand I of Austria’s parents were double first cousins, which implies that he only had 4 great-grandparents instead of 8, 8 great-great-grandparents instead of 16 and so on. Since everyone has some marriages among relatives in their ancestry, their cumulative effect reduces the expected number of ancestors at generation n from 2^n to an unknown number, unique to each individual. The overall shape of an ascendant family tree, if presented with the oldest generations at the top, should therefore not look like an inverted pyramid but rather like a diamond (it should get wider and wider but eventually start to narrow down to few ancestors). The aim of this paper is to present a mathematical model that helps to study the influence of inbreeding on the number of ancestors in a family tree, hence on its shape. Considering that the reader might not be familiar with the mathematical jargon, this paper focuses on the main ideas and results and does not go into details. For example, we avoid writing any formulas as much as possible. For the more expert reader, however, the endnotes provide the appropriate technical terms needed to fully understand the model and its properties. The model output is the ascendant family tree of one individual. Since our goal is to study the influence of inbreeding, the model parameters are linked to the proportions of cousin unions at each generation in the studied population or community. Family trees may vary greatly in size and complexity, even within a single community. This is why we propose a stochastic model. This means that the output is random; for the given parameters, the family tree may be different each time. A deterministic model on the other hand, would consistently provide the same tree. “Random” does not mean “any” family tree. The randomness is controlled by our model, in the sense that some family trees are more likely to be obtained than others. For example, if we choose as parameters large proportions of first cousin unions, the output very likely will be a rather narrow family tree (with few ancestors at each generation), and with small probability, be a very large tree.

Mots clés

Fichier principal
Vignette du fichier
PenissonGenealogySciences(2020).pdf (731.47 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Licence

Dates et versions

hal-04272628 , version 1 (07-11-2023)

Licence

Identifiants

  • HAL Id : hal-04272628 , version 1

Citer

Sophie Pénisson. A Mathematical Model for the Ancestor Paradox. Proceedings of the Symposium on Genealogy and the Sciences, Dec 2018, Rehovot, Israel. pp.112-119. ⟨hal-04272628⟩
246 Consultations
714 Téléchargements

Partager

  • More