Preferred attachment in affiliation networks

Vertices of an affiliation network are linked to attributes and two vertices are declared adjacent whenever they share a common attribute. For example, two customers of an internet shop (or video-sharing website) are called adjacent if they have purchased (or downloaded) the same or similar items. Assuming that each newly arrived customer is linked preferentially to already popular items we obtain a preferred attachment affiliation network that evolves in time. We show that the fraction of customers having i neighbours scales as i−2−α ln i for large i. Here α > 0 is the ratio between the two intensities: intensity of the flow of customers and that of the newly arriving items.