Depths in hooking networks

Abstract

Abstract A hooking network is built by stringing together components randomly chosen from a set of building blocks (graphs with hooks). The vertices are endowed with “affinities” which dictate the attachment mechanism. We study the distance from the master hook to a node in the network chosen according to its affinity after many steps of growth. Such a distance is commonly called the depth of the chosen node. We present an exact average result and a rather general central limit theorem for the depth. The affinity model covers a wide range of attachment mechanisms, such as uniform attachment and preferential attachment, among others. Naturally, the limiting normal distribution is parametrized by the structure of the building blocks and their probabilities. We also take the point of view of a visitor uninformed about the affinity mechanism by which the network is built. To explore the network, such a visitor chooses the nodes uniformly at random. We show that the distance distribution under such a uniform choice is similar to the one under random choice according to affinities.