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.

Caching data together with expiration times beyond which the data are no longer valid is a standard method for promoting information coherence in distributed systems, including the Internet, the World Wide Web (WWW), and Peer-to-Peer (P2P) networks. We use the framework of competitive analysis of online algorithms and study upper and lower bounds for page eviction strategies in the case where data have expiration times. We show that minimal adaptations of marking algorithms achieve performance similar to that of the well-studied case of caching without the expiration time constraint. Marking algorithms include the widely-used Least Recently Used (LRU) eviction policy. In practice, when data have expiration times, the LRU eviction policy is used widely, often without any consideration of expiration times. Our results explain and justify this standard practice.

The connectivity of the autonomous systems (ASs) in the Internet can be modeled as a time-evolving random graph, whose nodes represent ASs (or routers), and whose edges represent direct connections between them. Even though this graph has some random aspects, its properties show it to be fundamentally different from "traditional'' random graphs. In the first part of this paper, we use real BGP data to study some properties of the AS connectivity graph and its evolution in time. In the second part, we build a simple model that is inspired by observations made in the first part, and we discuss simulations of this model.