Connectivity Transitions in Networks with Super-Linear Preferential Attachment

We analyze an evolving network model of Krapivsky and Redner in which new nodes arrive sequentially, each connecting to a previously existing node _b_ with probability proportional to the _p_th power of the in-degree of _b_. We restrict to the super-linear case _p_ > 1. When , the structure of the final countable tree is determined. There is a finite tree T with distinguished _v_ (which has a limiting distribution) on which is "glued" a specific infinite tree; _v_ has an infinite number of children, an infinite number of which have _k_ - 1 children, and there are only a finite number of nodes (possibly only _v_) with _k_ or more children. Our basic technique is to embed the discrete process in a continuous time process using exponential random variables, a technique that has previously been employed in the study of balls-in-bins processes with feedback.