Krapivsky, P. L.,Redner, S.

We investigate choice-driven network growth. In this model, nodes are added one by one according to the following procedure: for each addition event a set of target nodes is selected, each according to linear preferential attachment, and a new node attaches to the target with the highest degree. Depending on precise details of the attachment rule, the resulting networks have three possible outcomes: (i) a non-universal power-law degree distribution; (ii) a single macroscopic hub (a node whose degree is of the order of N, the number of network nodes), while the remainder of the nodes comprise a non-universal power-law degree distribution ;(iii) a degree distribution that decays as (k ln k)(-2) at the transition between cases (i) and (ii). These properties are robust when attachment occurs to the highest degree node from at least two targets. When attachment is made to a target whose degree is not the highest, the degree distribution has the ultra-narrow double-exponential form exp(-const. Chi e(k)), from which the largest degree grows only as ln ln N.