## Internet Mathematics

- Internet Math.
- Volume 2, Number 2 (2005), 121-163.

### Connectivity Transitions in Networks with Super-Linear Preferential Attachment

Roberto Oliveira and Joel Spencer

#### 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.

#### Article information

**Source**

Internet Math., Volume 2, Number 2 (2005), 121-163.

**Dates**

First available in Project Euclid: 16 January 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.im/1137446619

**Mathematical Reviews number (MathSciNet)**

MR2193157

**Zentralblatt MATH identifier**

1097.68016

#### Citation

Oliveira, Roberto; Spencer, Joel. Connectivity Transitions in Networks with Super-Linear Preferential Attachment. Internet Math. 2 (2005), no. 2, 121--163. https://projecteuclid.org/euclid.im/1137446619