Exponential Growth and Decay Model for Human Genealogy (Common Ancestor)

Question

First time poster, and, as my post will intimate, not a mathematician, just someone searching for answers. My question has two parts:

1) In a genealogical chart for a single individual (called an Ahnentafel) starting with yourself and working backwards, you'll find a simple exponential trait to your preceding/antecedent group of ancestors, e.g.

• you have one set of parents (2 people)
• you have two sets of grandparents (4 people)
• you have four sets of great-grandparents (8 people)
• and so on...I'm only counting genetically-linked lineage (no step/half) for simplicity and using "sets" of ancestors rather than individuals.

However, whether you believe in Adam and Eve or in Darwin and Haldane, at a certain point all of this must converge back to an original set of antecedents (your common, original male/female ancestors, and logically the common human ancestors for all--the question I'll leave to philosophers and Richard Dawkins is how you get to a single ancestor not a single set of ancestors). Again, for simplicity, I'm only counting homo homo sapiens and not trying to take this back to the first unicellular organisms.

The question I'm trying to answer is, as one moves back in time, away from yourself (x=1) on a genealogical chart, your ancestors increase exponentially, but at some point they must start decreasing again to get back to a single set of common original ancestors (y=1)--for argument's sake, let's assume the decrease is perfectly proportional to the rate of increase and the time-series is based on finite generations not years--though if someone wants to try and model out interbreeding have at it.

When would this conversion/inflection across the generations need to occur--put another way, what is the maximum number of grand-nth parent sets you'd need to have before we started to see a need for this decrease--one would imagine it's about half-way back? In highly simple form it would go 1:2:4:2:1, but on a much grander scale.

There's an excellent article here from the BBC that talks about this issue as well as one known as the "genealogical paradox" (i.e. that most genealogy models show one to have more potential ancestors than human beings to have ever lived), and it also provides an important parameter for the time series: human history back to a single set of common ancestors for all humans is only about 3000 years or 100 generations. It also points out the need to assume inbreeding, consanguity, and incest as part of any genealogy, but for reasons both moral and mathematical let us keep things pure and simple.

(Note: the progression back 100 generations without assuming inbreeding would show more than a trillion (maybe even quadrillion or quintillion) potential ancestors and most estimates show only 100 billion people have EVER lived on Earth...here's an article on the "diamond-shaped theory of ancestors" and another on what's called "pedigree collapse"

There's also an excellent previous question in a similar vein that can be found here and provides some further helpful terms and guidance: Mathematics of genealogical trees

2) The second part of my question pertains to the first: formulaically, how would one model out the math for the specific question above using the parameters described (e.g. 100 generations)? And, more generally, how would one write the formula for an exponential growth time series that starts at 1 and that must then suddenly inflect, and start to decay in proportion to its original exponential growth to ensure the final result is 1 at the end of the sequence? Put another way, what is the general formula for expressing a pattern that both increases and decreases across a time series such as 1:2:4:2:1 and could this be expressed in a single formula?

For bonus points: what fields of math are we discussing in this question and what would the graph for the specific ancestor and general formula equation look like? I believe in graph theory this is something called a directed acrylic graph?

Thank you all!

Answer 1

For a simple model you need the population of the earth as a function of time. You then assume the ancestors are randomly drawn from the population. This overstates how far back you have to go to get collisions in the tree because your ancestors came from one or a few locations and people far away from there had no chance to be your ancestor.

I am 65 and my parents were born about 1930. If we take a generation time of 25 years, by the same logic as the birthday problem we expect the first collision about when the number of ancestors is the square root of the world population. The world had a billion people about 1800, but I had only 64 ancestors then. Each century multiplies my number of ancestors by 16, so in 1700 I had 1024, in 1600 I had 16k which squares to 256M while the population was about half a billion. The first collision then occurred somewhere around then, probably later because of localization. That is only 14 generations ago. If you draw my ancestors randomly with replacement from the population, the number of times an individual shows up is a Poisson distribution. Another 16 generations, 400 years, and the number of ancestors is about the world population, so most people in the world then are my ancestor.

Answer 2

I was thinking about this around the same time you posted and found a paper I had written.
Think of a family tree, on the root it's you, then your parents, grandparents and so on. Many of the nodes have to represent the same person, because else one generation (level) of your tree would be extremely large.

I found a formula, which I present below.

a person $p$ is represented by $n_p(l)$ nodes on each level $l$ on the tree (is an ancestor to you from $\sum\limits_{l} n_p(l)$ different families)

Now, consider the person who is an ancestor to you from more than 1 family and doesn't have a descendant who is an ancestor to you from more than 1 family.
To get the true population of our ancestors each time we take out the subtrees (family) with roots those people, expect for one (one on the highest level).

Below, $l$ stands for level and $P_n$ stands for the number of people on the $n$-th level (generation). I define $\lambda_l=\sum\limits_{p\in\text{people}} \{\ n_p(l)-[is\_high_p(l) ] \ \} ,\quad$ the summand is non-zero only for the people I described$\ \ $

and $ \ [is\_high_p(l) ]\ $equals $1$ if the person $p$ is represented by some node on level $l$ and from none on higher levels, otherwise $0$. Then:

$P_n=2^n-\lambda_1 2^{n-1}-\lambda_22^{n-2}-\dots-\lambda_n$
($ \ \ $ notice that $P_{n+1}=2P_n-\lambda_{n+1}\ \ $.. $\lambda_{n+1}$ is the amount of nodes we erase on level $n+1$ )
and the number of people on your family tree is $\sum\limits_{i=0}^{n}P_i \ = (2^{n+1}-1)-\lambda_1(2^{n}-1) -\lambda_2(2^{n-1}-1)-\dots-\lambda_n(2^{1}-1) $

$\begin{equation} \left( \begin{array}{} \text{I started from this equation: $\ P(n,i)=(1-\frac{\lambda_i}{P(i,i-1)}).P(n,i-1)\ ,\quad P(n,0)=2^n$} \\ \text{$P(n,i)$ be the number of people in the n-th generation after the $i$-th consideration (taking out subtrees of the} \\ \text{$i$-th level) and we are interested in $\ P_n=P(n,n)$.} \end{array} \right) \end{equation}$

For the generations' people to suddenly be less as we go up the tree, what I think you asked: $P_{n+1}>P_{n}\ \iff \ \ (\lambda_2-\lambda_1)2^{n-1}+(\lambda_3-\lambda_2)2^{n-2}+\dots+(\lambda_{n+1}-\lambda_n)\ \geq \ \ 2^n$
$\lambda_1=0$ because the parents can't be the same person (you're not cloned)
$\lambda_2\in \{0,1\}$
$\text{if}\ (\lambda_2=0) \ \lambda_3\leq 3\ \ \ , \ \ \text{if}\ (\lambda_2=1)\ \lambda_3\leq 2$
a first guess (most probably false): $\lambda_{n+1}\in\{0,\dots,2^{n}-2^{\lambda_n}\}$.

In reality,of course you can learn more, using this as the base. You can use time and other facts to restrict the values of $\lambda$ further, For example,
1.let's say two parents can't differ more than 90 years in age.
2.Imagine nodes of one branch of your family might have reproduced at very young ages and the other at very old ages, but that mathematically limits the number of people represented in both families by nodes.
3.statistics or scientific discoveries and history.
, so maybe the same person can't be represented by nodes on too many levels and other restrictions hold.