Connectivity Transitions in Networks with Super-Linear Preferential Attachment

Abstract

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 $1+\frac{1}{k} < p < 1 + \frac{1}{k-1}$, the structure of the final countable tree is determined. There is a finite tree $\mbox{T}$ 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.