View from the Pennines: Making Connections

Our village prides itself in its isolation: visitors can buy T-shirts showing the name of the village, a cartoon sheep, and the words Back O’Beyond. Even so I can find a connection from myself to my mother through the village involving only three steps (i.e. I know someone in the village who knows someone who knows my mother). Since my mother moves in London literary circles I find this remarkably short. Trying a different tack, I wondered how far I am from having played football with David Beckham. Again the answer (I have played football with someone who has played football with...) seems surprisingly low: four. These are examples of the ‘small world’ phenomenon that social scientists have been studying for over forty years: social connections lead to remarkably short average path lengths between individuals. The last five years has seen mathematics beginning to make headway in explaining this empirical observation (see the recent books [6, 15]). Mathematically a social network, and many other networks, can be thought of as a graph: it has n vertices representing people, with edges connecting two people if they are friends. It is usual to ignore the unrequited angst of non-reflexive friendship and so the edges are not directed, although such complications can be added. Thus there are at most M = 12n(n − 1) edges since each of n vertices can be connected to at most n − 1 other vertices, and so there are n(n−1) ordered pairs of vertices which could have edges, and the