In this paper we focus on the problem of the degree sequence for a random graph process with edge deletion. We prove that, while a specific parameter varies, the limit degree distribution of the model exhibits critical phenomenon.

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area) (T) covered by the process in the time interval [0,T]. The Laplace transform of (T) follows as a consequence. The main proof involves a double Laplace transform of (T) and is based on excursion theory and local time for Brownian motion.

We propose the following simple stochastic model for phylogenetic trees. New types are born and die according to a birth and death chain. At each birth we associate a fitness to the new type sampled from a fixed distribution. At each death the type with the smallest fitness is killed. We show that if the birth (i.e. mutation) rate is subcritical, we obtain a phylogenetic tree consistent with an influenza tree (few types at any given time and one dominating type lasting a long time). When the birth rate is supercritical, we obtain a phylogenetic tree consistent with an HIV tree (many types at any given time, none lasting very long).