Properties of a random attachment growing network

In this study we introduce and analyze the statistical structural properties of a model of growing networks which may be relevant to social networks. At each step a new node is added which selects k possible partners from the existing network and joins them with probability delta by undirected edges. The "activity" of the node ends here; it will get new partners only if it is selected by a newcomer. The model produces an infinite-order phase transition when a giant component appears at a specific value of delta, which depends on k. The average component size is discontinuous at the transition. In contrast, the network behaves significantly different for k=1. There is no giant component formed for any delta and thus in this sense there is no phase transition. However, the average component size diverges for delta> or =1/2.