Universal features of surname distribution in a subsample of a growing population

TL;DR

This paper investigates surname distribution in a growing population, deriving theoretical models for family size statistics and applying them to U.S. Census data, revealing significant discrepancies in growth rate estimates.

Contribution

It introduces a novel theoretical framework for analyzing family size distributions in subsamples of growing populations, accounting for mutation and growth dynamics.

Findings

Derived the family size distribution for subsamples from the full population distribution.

Showed that subsample distribution shifts left, affecting family size statistics.

Applied the model to U.S. Census data, revealing a misestimation of population growth rate.

Abstract

We examine the problem of family size statistics (the number of individuals carrying the same surname, or the same DNA sequence) in a given size subsample of an exponentially growing population. We approach the problem from two directions. In the first, we construct the family size distribution for the subsample from the stable distribution for the full population. This latter distribution is calculated for an arbitrary growth process in the limit of slow growth, and is seen to depend only on the average and variance of the number of children per individual, as well as the mutation rate. The distribution for the subsample is shifted left with respect to the original distribution, tending to eliminate the part of the original distribution reflecting the small families, and thus increasing the mean family size. From the subsample distribution, various bulk quantities such as the average family size and the percentage of singleton families are calculated. In the second approach, we study the past time development of these bulk quantities, deriving the statistics of the genealogical tree of the subsample. This approach reproduces that of the first when the current statistics of the subsample is considered. The surname distribution from th e 2000 U.S. Census is examined in light of these findings, and found to misrepresent the population growth rate by a factor of 1000.