The city as a giant component: a random graph approach to Zipf's law

The emergence of a 'city' out of a set of locations in space can be considered akin to the evolution of a random graph. Interaction between individuals who are connected to each other is at the source of the benefits associated with a city. If the interaction probability rises, a threshold is eventually crossed at which point most of the graph becomes connected, giving rise to a grand component. It is at this point that a viable 'city' emerges. This view suggests an interpretation of Zipf's law, which we test using US Census data.