Yule (preferential attachment process vs. birth&death process What is the distinction between the Yule (preferential attachment) process and the birth&death process? Are they the same thing, called differently in different contexts, or is the birth&death a special case of the Yule process?
 A: Yule process is the same as the pure birth process, although the two names are apparently used in somewhat different contexts.
One speaks of birth&death process when it is described in terms of differential equations and Markovian dynamics, as, e.g., here. The pure birth process then would be usually described by a set of rate equations
$$
\dot{P_n}(t) = \lambda (n-1)P_{n-1}(t) - \lambda n P_n(t),
$$
where $P_n$ is the probability of having $n$ specimen, and the probability of the birth of a new speciman is proportional to the number of the already existing ones, $\lambda n$.
Name Yule process is more common in the context of the survival analysis (more precisely - the analysis of recurrent events), where the non-homogeneous birth process is defined via its intensity as (the following is actually taken from the French version):
$$
\forall t \in \mathbb{R}_+, \forall j \in \mathbb{N} : \\
\begin{cases}
N(0) = 0\\
E(dN(t)|\mathcal{N}_{t-})=E(dN(t)|N(t-)=j)=\alpha(j)\lambda(t)dt
\end{cases}
$$
One then refers to case $\alpha(j)=j$ as Yule process, and $\alpha(j) = 1 + \alpha j$ as linearly extended Yule process (LEYP).
