Are you and I by any chance related? According to a report by Bernard Derrida of the Ecole Normale Superieure, Paris, France and colleagues in Physical Review Letters [1 March 1999], the statistical properties of genealogical trees suggest that we could very well share a distant ancestor.
Genealogies pose an apparent paradox. Each of us is the offspring of two parents, who likewise both had a biological mother and father. This means we have two parent, four grandparents, eight great-grandparents and so on. In other words, our ancestors increase exponentially the further back we look. About 20 generations (about 400 years), ago we each have about a million ancestors - and after that the numbers start to get even sillier. Forty generations ago (800 years) gives us one trillion ancestors, and fifty gives one quadrillion. This is not only many, many more people than live on the planet today - it is many more than have ever lived.
So what's wrong with this calculation? Simple: the genealogy will inevitably contain some people more than once. This happens whenever two siblings appear in your personal tree. Then only one couple, not two couples, is needed to generate the two siblings. In other words, the appearance of siblings duplicates entire branches of the tree, because each sibling lies at the apex of exactly the same earlier genealogy. Our genealogical tree with astronomical numbers of individuals therefore contains many, many repeats - the number of different people it contains is much smaller.
But why should two siblings appear in your tree at all, if you're tracing only your parents, their parents, and so forth, and ignoring each of their siblings? Because of mating within the family. If first cousins pair up and produce a child, that child's grandparents were siblings.
This resolution of the genealogy paradox was explained in 1996 by Susumu Ohno of the Beckman Research Institute of the City of Hope in Duarte, California. What Derrida's team have now done is to use considerations of this sort to look at the statistical properties of genealogies, and in particular to examine the chances of you and the woman who sold you your train ticket this morning sharing a common ancestor.
In the modern world, with its shifting demography and widespread immigration, that is a very difficult question to answer. But things were not always so complicated: once populations were far less mobile, communities were more fixed and stable. Derrida and colleagues have looked at situations where the issue of genealogy is more constrained.
One such is the family tree of the English royalty, exemplified by Edward the Third (1312-1377). Royal genealogies are manageable because, particularly in the past, those of noble birth married only other nobles. Because there were perhaps only a few thousand partners to choose from in Edward III's time, his genealogy reveals considerable interbreeding - evinced by the frequent repetition of individuals in the genealogical tree. (Repetitions, remember, indicate a close family mating, such as first cousins, in the past.) Some individuals appear up to six times in Edward III's tree.
Derrida and colleagues show that the chance of repetitions in this royal ancestry can be predicted by a model of a population in which each generation is the same size as the last, and each person is assigned two parents at random from the previous generation. This sounds like a strange model, perhaps; but it is in fact not so unrealistic a description of a community whose size stays the same, since everyone's parents must be found somewhere amongst the previous generation.
The researchers explored the statistics of the genealogical trees produced by this model population after many generations. They found that about 80 percent of the entire population of the community appear in the tree of any one individual when traced back far enough.
This means that the chance that two people from a given generation share a common ancestor in the distant past is about 64 percent. That remains true no matter how large the population, provided that its size remains the same through time.
Of course, many real communities today are permeated by individuals whose parents were not a part of that community - so the model can't be relied on to describe this case. And most real communities, of humans at least, increase in size as time passes. But all the same, it suggests that you might be more closely related than you think to those strangers on the street.