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2 Box 1 Graph-theoretical definitions The length of a path in a graph, G, is the number of edges in the path. For each pair of nodes i and j in G, the distance d(i, j) is defined to be the length of a shortest path joining i and j. The radius of G is R ¼ min{ max dði;kþ} i[g k[g and a node i is called a centre of G if max k[g d(i, k) ¼ R. Assume R \$ 1; the case R ¼ 0(G has one node) was treated previously 2. For each centre node i, let S i be a set of minimal size that consists of neighbours of node i and satisfies min {d( j, k): j [ {i} < S i } # R 2 1 for all k [ G. H i is defined as the number of nodes in S i, H is the minimum of H i over all centres i, and D ¼ H : The diameter of G is D ¼ max i,k[g d(i, k). exclusively maternal lines would have lived something like 50,000 times earlier in the order of one million generations ago. As genealogical ancestry is traced back beyond the MRCA, a growing percentage of people in earlier generations are revealed to be common ancestors of the present-day population. Tracing further back in time, there was a threshold, let us say U n generations ago, before which ancestry of the present-day population was an all or nothing affair. That is, each individual living at least U n generations ago was either a common ancestor of all of today s humans or an ancestor of no human alive today. Thus, among all individuals living at least U n generations ago, each present-day human has exactly the same set of ancestors. We refer to this point in time as the identical ancestors (IA) point. As with the MRCA point, the IA point is also quite recent in a randomly mating population: U n, 1.77 log 2 n generations ago 2. The major problem in applying these results to human populations is that mating is not random in the real world. Mating patterns are structured by geography, proximity, culture, language and social class. Nevertheless, even in populations with considerable internal structure, the time to the MRCA can be remarkably brief. To demonstrate this in a tractable mathematical model, consider a population of size n divided into randomly mating subpopulations that are linked by occasional migrants. The population is represented by a graph, G, with a node for each subpopulation. Edges indicate pairs of nodes that exchange a small number (for example, one pair) of migrants per generation. Let R denote the radius of G, and let D be a quantity ranging between 0 and 1 that depends on the structure of G (see Box 1). A probabilistic analysis (see Supplementary Information) shows that as n!1, T n, ðr þ DÞ log 2 n: Furthermore, if we let D denote the diameter of the graph, then the number of generations, U n, since the IA point satisfies U n, ðd þ 1:77Þ log 2 n: Computer simulations accord with these theoretical predictions. Tables 1 and 2 give distributions of T n and U n for small populations of varying sizes in graphs with one node, three connected nodes, five fully connected nodes and for a ten-node graph loosely based on world geography as shown in Fig. 1. In these simulations, neighbouring subpopulations exchange one pair of migrants per generation. Each mean is calculated from 100 model runs. Although guaranteed to be accurate only for sufficiently large n, the theoretical predictions describe the simulations quite well even for models with just a few thousand individuals. Whenever n is doubled, T n is expected to increase by R þ D, and U n is expected to increase by D þ These predicted increases, which are listed in the last columns of Tables 1 and 2, agree closely with the simulation results. To hazard a rough first guess about human recent common ancestors, we could extrapolate the results for the graph of Fig. 1 to a growing population with a final size of 250 million. When applying this model to a growing population, the fixed population size that provides the best approximation is the size at the time that the MRCA lived. We take this effective population size to be 250 million, which is approximately the global population in the year AD 1. Starting from n ¼ 16,000, a population of 250 million is reached by doubling 14 times. Approximating the increases in T n and U n beyond the values seen in Tables 1 and 2 by their theoretical predictions for each doubling of n, we arrive at T n < 34 þ 14 3 ¼ 76 generations (about 2,300 years) and U n < 74 þ 14 6:77 ¼ 169 generations (about 5,000 years). These estimates would suggest, with the exchange of just one pair of migrants per generation between large panmictic populations of realistic size, that the MRCA appears in about the year 300 BC, and all modern individuals have identical ancestors by about 3,000 BC. Such estimates are extremely tentative, and the model contains several obvious sources of error, as it was motivated more by considerations of theoretical insight and tractability than by realism. Its main message is that substantial forms of population subdivision can still be compatible with very recent common ancestors. The dynamics of human subpopulations are much more complex than those in the simple graph model discussed above. Although these complexities make theoretical analysis difficult, a computer model incorporating more complicated forms of population substructure and migration allows the demographic history of human populations to be simulated. The Supplementary Information contains more details on the model and computations; here we briefly outline some of the main points. This model is based on a simplified projection of the world s Table 1 Simulations of T n Graph n ¼ 1,000 n ¼ 2,000 n ¼ 4,000 n ¼ 8,000 n ¼ 16,000 R þ D One node 10.8 (0.4) 11.8 (0.4) 12.8 (0.4) 13.9 (0.3) 14.8 (0.4) 1.00 Three fully connected nodes 14.0 (0.7) 15.6 (0.7) 17.1 (0.9) 18.9 (0.8) 20.3 (1.0) 1.50 Five fully connected nodes 14.0 (0.5) 15.8 (0.5) 17.8 (0.5) 19.6 (0.5) 21.5 (0.6) 1.75 Ten-node graph shown in Fig (1.3) 24.3 (1.5) 27.6 (1.5) 30.5 (1.5) 33.8 (1.7) 3.00 Means (standard deviations in parentheses) of Tn (the number of generations back to the MRCA) for graph-structured populations exchanging a single pair of migrants per edge per generation. The last column shows R þ D, the expected asymptotic increase in Tn per doubling of n. Table 2 Simulations of U n Graph n ¼ 1,000 n ¼ 2,000 n ¼ 4,000 n ¼ 8,000 n ¼ 16,000 D þ 1.77 One node 20.8 (1.6) 22.6 (1.5) 24.6 (1.5) 26.5 (1.6) 28.3 (1.4) 1.77 Three fully connected nodes 27.4 (1.5) 30.3 (1.4) 33.4 (1.5) 36.2 (1.7) 38.9 (1.5) 2.77 Five fully connected nodes 25.9 (1.3) 28.9 (1.4) 32.1 (1.7) 35.3 (1.5) 37.9 (1.4) 2.77 Ten-node graph shown in Fig (2.7) 53.0 (2.7) 59.8 (2.7) 66.8 (2.9) 73.6 (2.7) 6.77 Means (standard deviations in parentheses) of U n (the number of generations back to the IA point) for graph-structured populations exchanging a single pair of migrants per edge per generation. The last column shows D þ 1.77, the expected asymptotic increase in U n per doubling of n Nature 563 Publishing Group

3 Figure 1 World map viewed as a ten-node graph. This graph has radius 3 and diameter 5. actual inhabited land masses and has three levels of substructure: continents, countries and towns. Figure 2 depicts the model s geography and migration routes used before AD 1500, with the countries shown as squares and the number of towns per country differing from continent to continent. Towns and countries represent both the local geographical areas and the relevant social and ethnic groups from which most people find mates. The model uses a simplified migration system in which each person has a single opportunity to migrate from his or her town of birth. The probabilities of leaving a town or a country are set at various levels to reflect different migration patterns. Migrants who move between towns can travel to any other town within the country. A migrant who leaves a country for another country within the same continent chooses the destination with a probability that diminishes as the inverse square of the geographical distance. Each continent has a number of port countries from which migrants can travel to another continent. A fixed, large percentage (for example, 95% in some simulations) of the migrants through a port come from the country in which the port is located, with the remainder drawn from other countries in the continent in proportion to their inverse squared distance. The value next to a port in Fig. 2 is its migration rate, in people per generation, and the date in parentheses indicates when the port opens, if it is more recent than the start of the simulation in 20,000 BC. When a port opens, there is usually a single generation of migration at a higher rate than the steady-state rate shown in the figure. After the year AD 1500, additional large ports, which are not shown, begin to open to simulate colonization of the Americas, Australia and elsewhere. Immediately before this, the native population of the Americas is markedly reduced to simulate the effects of European-introduced diseases 7. Generations overlap in this model and we explicitly simulated the lifespan and the times at which mating and reproduction events occur for each individual 8,9, as described in more detail in Supplementary Information. The birth rate of each continent or island was individually adjusted so that the populations match historical Figure 2 Geography and migration routes of the simulated model. Arrows denote ports and the adjacent numbers are their steady migration rates, in individuals per generation. If given, the date in parentheses indicates when the port opens. Upon opening, there is usually a first-wave migration burst at a higher rate, lasting one generation. 564

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