The basic simulation procedure is the same as previously [3], except that the population is undivided and reproduction is monogamous. To summarize, population size is constant and generations are non-overlapping. Each simulation begins at Generation 0 and proceeds forward for a predetermined number of generations. The only information recorded for individuals in subsequent generations is their Generation 0 ancestors. For most analyses, it was necessary only to have qualitative (0 or 1) information about ancestors. The exception was the analysis of quantitative genealogical overlap [4]. Sib mating was permitted and presumably occurred at the frequency expected by chance. Because reproduction is monogamous, shared ancestry must necessarily involve pairs of ancestors, or couples. For brevity and clarity, however, I will use phrases such as “at least one shared ancestor”, with the understanding that “ancestor” actually means “ancestor pair”. All pairwise genealogical comparisons were made by selecting 1000 random pairs of individuals from the population in each generation. The simulations that are shown in Figure 2 involved sampling single (“focal”) individuals from the population, and comparing the Generation 0 ancestors of that individual to the ancestors of all other members of the population. For those simulations, the sample size of “focal” individuals was 10% of the population or 500, whichever was less. Unless otherwise noted, all results are based on 100 replicate simulations for each population size.

On average, the most recent shared ancestor of random pairs of individuals lived slightly more than 0.5 log₂N generations previously (Table 1). There is some suggestion that the mean time to shared ancestry as a fraction of log₂N decreases with population size, although the effect is very small for the population sizes simulated.

The same set of simulations permits determination of the probability that random pairs of individuals will share at least one ancestor of specified degree. For all population sizes that were examined, the probability that two individuals will share an ancestor who lived 0.5 log₂N generations ago, or more recently, is approximately 0.5 (Figure 1). For N = 20,000, 0.5 log₂N is about 7. Thus, the probability that a random pair of individuals drawn from a population of 20,000 will share a 7^th-generation (or more recent) ancestor is about 50%. Seventh generation corresponds to fifth-great grandparent, and individuals who share such an ancestor are 6^th cousins. The probability of shared ancestry increases very rapidly if additional generations are considered: for ancestors who lived 0.6 log₂N generations in the past, or more recently, the probability of shared ancestry. is greater than 90% for all population sizes that were simulated; and the probability of shared ancestry reaches 100% if we include ancestors who lived about 0.7 log₂N generations previously. That is, there is universal pairwise shared ancestry: every individual in the population is related to every other individual by shared ancestors who lived 0.7 log₂N, or fewer, generations in the past.

An important feature of Figure 1 is that the time scale (x-axis) is not generations, but generations relative to log₂N. The fact that the curves lie on top of one another shows that the probability of pairwise shared ancestry scales uniformly with log₂N, albeit with some spread at the inflection points.

This analysis can be extended to consideration of shared ancestry in samples of S individuals. Consider a sample of S = 10 from a population N = 20,000. From above, the probability that any pair of individuals do not share an ancestor who lived 7 generations ago (or more recently) is approximately 0.5. For S = 10, there are 45 pairwise comparisons. Thus, the probability that a sample of 10 individuals contains no 6^th, or less distantly related, cousins is approximately 0.5⁴⁵, or about 2.8 × 10⁻¹⁴ (assuming independence). On the other hand, for N = 20,000, the probability that all individuals in a sample of any size will be related to one another as 9^th (or closer) cousins is very nearly 100%, as will be verified below.

A random individual will be related to other individuals in the population by shared ancestors of various degrees. This idea can also be expressed as: what proportion of the population will be an individual’s k^th degree, or closer, cousins? Results are shown in Figure 2. For example, a random individual will be related to about 50% of all other individuals in the population by shared ancestors who lived 0.5 log₂N generations previously, or more recently. If N = 20,000, those relatives are 6^th degree, or closer, cousins. Figure 2 is almost identical to Figure 1, even though they depict the results of independent, and procedurally different sets of simulations. However, that is to be expected. For example, if an individual has a 0.5 probability of being related to another individual drawn randomly from the population (Figure 1), then we expect that the same individual will be related by similar degree to about 50% of the population (Figure 2). In short, the y-axis labels of the two figures are interchangeable. Considering ancestors who lived 0.7 log₂N generations ago, or more recently, every individual in the population is related to every other individual (Figure 2), although not necessarily by the same ancestors—in other words, there is universal pairwise shared ancestry. Hence, the assertion in the previous section that all members of a sample of any size from a population of N = 20,000 will be related to each other by ancestors who lived 10 or fewer generations ago (10 ≈ 0.7 log₂ 20,000).

Pairwise qualitative genealogical overlap is the proportion of Generation 0 ancestors that are shared by pairs of individuals in subsequent generations. It is calculated as follows. Let individual A have N_A different ancestors from generation 0, and individual B, N_B different ancestors. Let A and B share N_AB ancestors from Generation 0. Then the qualitative genealogical overlap due to shared ancestors who lived G generations in the past is (N_AB / N_A + N_AB / N_B)/2. The range of values for this index is 0 - 1.0.

About 0.9 log₂N generations are required for 0.5 overlap (Figure 3). This is somewhat less than the time required for a population-wide MRCA. An overlap of 0.5 means that pairs of individuals share half of their Generation 0 ancestors, on average. By about 1.3 - 1.5 log₂N generations, qualitative pairwise overlap is >0.99: larger populations require less relative time. Qualitative overlap is necessarily 1.0 for all pairs when Generation 0 is an identical ancestry generation: the average time required for that is about 2 log₂N generations.

Quantitative overlap is the similarity in the frequencies of shared ancestors in the genealogies of a pair of present-day individuals. I use the metric q^{(α, β)}(G), introduced by Derrida et al. [4], which is the overlap between the trees of individuals α and β at generation G in the past. The range of values is 0 - 1. The largest population size simulated was 16,000, due to computer limitations.

Quantitative overlap (Figure 4) is almost indistinguishable from qualitative overlap (Figure 3). Overlap is about 0.5 for all population sizes for ancestors who lived sightly more than 0.9 log₂N generations in the past. Considering ancestors who lived about 1.4 - 1.6 log₂N generations previously, quantitative pairwise overlap is >0.99: larger populations require less relative time. Unlike the case of qualitative overlap, there is no requirement that quantitative overlap equal 1.0 once Generation 0 becomes an identical ancestry generation. But, in fact, quantitative overlap is >0.9999 by the time that identical ancestry occurs.

In these simulations, reproductive success (number of offspring) is Poisson distributed with mean 2.0. Extinction of Generation 0 lineages is rapid. In fact, the mean time to extinction is about 1.55 generations (independent of N), and the last extinction will occur by about 0.67 log₂N generations [3]. Consequently, about 80% of the of the original Generation 0 cohort will become persistent ancestors of the population in future generations [1,3,4]. A correlate of indefinite persistence is that each Generation 0 member eventually comprises a nearly fixed proportion of the ancestry of future generations. Different Generation 0 members will have different representations in future genealogies, but the distribution of those representations becomes approximately stationary [4,5].

The stationary distribution of the representation of Generation 0 members in the ancestry of future generations will be illustrated with an example from a single simulation with N = 16,000, and run for 30 generations. In this example, the last extinction of Generation 0 lineages occurred by Generation 8; 12,792 (79.95%) of the Generation 0 lineages persisted; the MRCA (12 Generation 0 individuals) appeared after 15 generations (=1.07 log₂N generations); and identical ancestry occurred after 27 generations (1.93 log₂N generations). A sample of 30 Generation 0 members was selected from this replicate for purposes of illustration. The lineages of 26 of those persisted, and all 26 became common ancestors of the population by Generation 22. Let C_Gij be the number of times Generation 0 member i occurs in the genealogy of individual j in Generation G. C_Gij is then summed over all individuals j in the population in Generation G, and scaled by dividing by 2^G. The resulting metric, C_Gi. is plotted for the sample of 26 persistent Generation 0 lineages from this replicate (Figure 5). Scaling by 1/2^G means that the expected value of C_Gi. is 1.0 for each Generation 0 member in each subsequent generation, and the sum of C_Gi. over all N Generation 0 members equals N in every generation.

The representation of each Generation 0 member in the ancestries of subsequent generations is initially highly variable, but then stabilizes fairly quickly. By about log₂N (approximately 14) generations, the distribution is very nearly static (Figure 5), and remains essentially unchanged for all future generations. The eventual shape of this distribution can be roughly inferred from Figure 5. A more detailed description is provided in Derrida et al. [4].

The preceding section illustrates the fact that the quantitative representation of each “successful” Generation 0 ancestor becomes approximately fixed, across generations, when summed over all individuals in the population. High values of pairwise quantitative overlap (Figure 4) would also seem to indicate that a given Generation 0 ancestor has very nearly equal representation in the genealogies of every individual within a generation. In other words, we might expect the scaled variance in the occurrence of a given Generation 0 member in the ancestries of individuals in later generations to become smaller with time. An appropriate statistic is the coefficient of variation (CV), defined as standard deviation/mean.

Clearly, any “successful” Generation 0 member must become a common ancestor of the entire population before it can have equal representation in the genealogies of all individuals in a future generation. In the replicate simulation described in the preceding section, about 72% of the eventual common ancestors had become so by Generation 18, and almost 98% by Generation 21. But even before that, most individuals in the population will be descendants of most Generation 0 members whose lineages have not gone extinct. Consider that with stable population size and Poisson-distributed reproduction, successful reproducers will have about 2.3 offspring, on average. After 15 generations, an “average” Generation 0 member will have 2.3¹⁵ (= 266,635) “descendants”. If N = 16,000, we might expect that most of the population will be included among those descendants.

A sample of 10% or 200, whichever was greater, of Generation 0 ancestors was used for calculation of the coefficient of variation. The CV was obtained for each Generation 0 lineage from the variance of the scaled C_Gij , the mean of which was C_Gi., as described in the previous section; and the CV then averaged over all persistent lineages in the sample of Generation 0 ancestors. Within about 1.7 log₂N generations or less, the CV declined to ≤0.10 (Figure 6). In fact, for N = 10,000 or 20,000, the CV was less than 0.05. In other words, the quantitative Generation 0 ancestry was very similar for all members of the population. That is consistent with pairwise quantitative overlap > 0.999 by this time for the three larger population sizes (Figure 4); and is to be expected given that each persistent Generation 0 lineage eventually represents a temporally stable portion of the ancestry of the whole population (Figure 5).