Distance-dependent connectivity: Yet Another Approach to the Small World Phenomenon

We investigate a relationship network of humans located in a metric space where relationships are drawn according to a distance-dependent probability density. The obtained spatial graph allows us to calculate the average separation of people in a very simple manner. The acquired results agree with the well-known presence of the small-world phenomenon in human relationships. They indicate that this feature can be understood merely as a consequence of the probability composition. We also examine how this phenomenon evolves with the development of human society.