Geekspeak: Why Life + Mathematics = Happiness. Graham Tattersall
there are probably around 16 members in your extended family. Now imagine going back two or three generations in Britain. A century ago, most couples had four or five children who survived into adulthood. It makes the table look dramatically different:
| Relative | Number of people |
|---|---|
| Your mother’s parents | 2 |
| Your father’s parents | 2 |
| Mother and her siblings | 5 |
| Father and his siblings | 5 |
| Uncles’/aunts’ partners | 8 |
| Cousins | 48 |
| Your siblings | 4 |
| Total | 74 |
Evidently, in societies with large families it’s the cousins who multiply almost exponentially with family size, and it is they who most likely provide the connectivity that holds communities together.
The reduction in the number of children in each family has had a profound and mathematically inevitable effect on community. In a small town of 10,000 inhabitants at the end of the nineteenth century, only a few hundred extended families would account for the entire population. You, or someone in your extended family, would be related to every other person in town.
Nowadays, with just 16 members in an extended family, you would have to live in a village of fewer than 200 geographically and socially immobile inhabitants to get the same level of coverage.
The tangible result of smaller families is low social connectivity and loss of community. Identity cards with biometric information and links to databases with our health records, past addresses and benefit history may be the inevitable synthetic replacement for first-hand knowledge about our neighbours.
A few years back, the Department of Trade and Industry organised a series of conferences on fraud prevention and security. They centred on the use of biometrics – technologies that use computer recognition of the iris, fingerprints or speech to uniquely identify an individual. One of those techniques is likely to be incorporated into British identity cards if they are ever introduced.
Grey suit after grey suit delivered the same technological gospel. Then, one speaker stood up and delivered a totally unexpected scathing attack on the whole idea. His argument: we have worked hard for our anonymity. It should be cherished as a prize of our development into a society based on individuals living without family. Who wants to go back to a village society where your every move is known to all?
Anyway, let’s get back to analysing your connection to Osama bin Laden or the Queen. We’ve established that a typical extended family has 16 members, or 15 other people you know very well. How about work associates?
Assuming that they are people you know by first name, the number will depend strongly on the kind of organisation you work in. Self-employed people might know just half a dozen customers by first name, whereas someone working in a large insurance office might know 40 or 50. We could take a typical number as 25.
And then there are your friends: most friends are known to you by their first name. You put in a figure for yourself. I think I have about 20 friends who are not anything to do with my work.
So far, the total number of people with whom you have a sufficiently close relationship to call them by their first name stands at about 60. That figure doesn’t include all the casual acquaintances whose name you might know: the man at the post office, the window cleaner, the man you meet walking his dog. Let’s say that makes another 20.
The total number of people with whom you are on first-name terms now stands at 80. Each of those people in your social circle is at the centre of their own personal social circle of about 80 people, and each of those people is at the centre of their circle. To simplify things, I’ll use ‘pal’ to denote anyone who is a friend, a relative, a work colleague or an acquaintance.
As a start, we can say that the number of pals of pals we have is about 80 × 80 = 6,400. But in real life some of the pals’ pals will be members of your own family, work colleagues and circle of friends, and the figure of 6,400 has counted those people twice over.
A simple way to take that overlap into account is to introduce a factor that expresses the percentage of each successive social circle that is common to the previous group. A factor of 50% seems fair. It means that, of the 80 people in the social circle of one of your pals, 40 will also be in your own circle.
So now, very conservatively, each link in the chain of social circles multiplies the number of people connected to you by a factor of just 40, but successive multiplication builds up the numbers very quickly. Try it out for six links in the chain:
| Links | Number of people |
| 1 | 40 |
| 2 | 40 × 40 = 1,600 |
| 3 | 40 × 40 × 40 = 64,000 |
| 4 | 40 × 40 × 40 × 40 = 2,560,000 |
| 5 | 40 × 40 × 40 × 40 × 40 = 102,400,000 |
| 6 | 40 × 40 × 40 × 40 × 40 × 40 = 4,096,000,000 |
The table shows there are a total of 4.1 billion people accessible through a chain of just six links. The Earth’s population is about 6.6 billion, so it seems that a pal of a pal of a pal of a pal of a pal will connect you to three-quarters of humanity.
Does it really work for anyone? Finding a link to Osama might be tricky, although certainly of interest to MI6. Bin Laden is a Saudi citizen, so a good bet would be to look for a link between yourself and someone in Saudi Arabia. Once you have found that link into the country, there are likely to be lots of links between your Anglo-Saudi bridge person and someone in bin Laden’s family.
The Osama example shows how the small number of links could be exploited by anyone using information about phone calls or emails between individuals. The first step is to construct a database of all known names. Listed under each name are the numbers for all incoming and outgoing phone calls or emails for that individual. That’s difficult, but technologically possible. Those phone numbers and email addresses are like the social circle of the named individual. A computer searches for chains with the same links.
We’ve already shown by simple arithmetic that there are probably no more than six links in the chain and that the people in the chain form a community of some kind. It may be benign or malign, but significantly it can be detected automatically by a computer.
It only takes one suspicious individual in the chain to cast suspicion on all the other people in the chain – the Queen, perhaps?
SPEAK GEEK
COUNTING FRIENDS OF FRIENDS, THERE ARE OVER A THOUSAND PEOPLE IN YOUR SOCIAL CIRCLE.
A typical person has 30 to 50 people they consider to be close friends or acquaintances. We could take an average of 40 for the purpose of calculation. If each of these friends has a further 40 friends, excluding you