How random is it when you unexpectedly run into someone you know? Chance encounters are less mysterious than they may seem. With a little mathematics, it is possible to predict how often people will meet, and even how often they narrowly miss each other.
We have all said it: “What a coincidence, running into you here!” But such encounters are not quite as unpredictable as they feel. If you know how fast people are walking, the size of the area in which they move, and the distance at which they would notice or recognise each other, you can build a mathematical model that predicts how often their paths will cross. Sandjai Bhulai explains this in a video of the Universiteit van Nederland.
A formula is one thing, but does it work when real people start walking around a real city? To find out, Universiteit van Nederland sent six colleagues into a defined part of the city for one hour. At every intersection, they were free to decide whether to turn left, turn right, or continue straight ahead. Their movements were followed from a control room while the experiment unfolded. Before the walk began, the mathematical model made a clear prediction: six encounters and nine near-encounters on average. Whether reality agreed with the mathematics is revealed in the episode.
The idea behind the calculation has its roots in mathematical models of moving individuals. Rather than trying to predict the precise route of every person, the model describes how frequently two independently moving people are expected to come sufficiently close to one another. Earlier research on mobile networks showed that, under suitable conditions, such meetings can be modelled surprisingly well as a random process, with the meeting frequency depending on factors such as speed, area and interaction distance.
What starts as a playful question about bumping into an acquaintance therefore connects to much broader problems. Similar mathematics can help researchers understand how infections spread through contacts, how people move through busy urban spaces or stations, and how search strategies can be organised efficiently. This highlights the connection between apparently mathematical puzzles and applications with societal relevance.
And the model leads to a slightly counterintuitive thought: the person you happen to meet today may be someone you have already narrowly missed many times without ever noticing.