An algorithm to represent inbreeding trees

Abstract

Recent work has proven the existence of extreme inbreeding in a European ancestry sample taken from the contemporary UK population (Yengo et al., 2019). This result brings our attention again to a math problem related to inbreeding family trees and diversity. Groups with a finite number of individuals could give a variety of genetic relationships. In previous works (Caruso and Jarne, 2015; Caruso and Jarne, 2014; Jarne and Caruso, 2019), we have addressed the issue of building inbreeding trees for biparental reproduction using Markovian models. Here, we extend these studies by presenting an algorithm to generate and represent inbreeding trees with no overlapping generations. We explicitly assume a two-gender reproductory scheme, and we pay particular attention to the links between nodes. We show that even for a simple case with a relatively small number of nodes in the tree, there are a large number of possible ways to rearrange the links between generations. We present an open-source python code to generate the tree graph, the adjacency matrix, and the histogram of the links for each different tree representation. We show how this mapping reflects the difference between tree realizations, and how valuable information may be extracted upon inspection of these matrices. The algorithm includes a feature to average several tree realizations, obtain the connectivity distribution, and calculate the mean value and standard deviation. We used this feature to compare trees with a different number of generations and nodes. The code presented here, available in Git-Hub, may be easily modified to be applied to other areas of interest involving connections between individuals, extend the study to add more characteristics of the different nodes, etc.