  The following random graph model was introduced for the evolution of
protein-protein interaction networks: Let $\mathcal G = (G_n)_{n=n_0,
n_0+1,...}$ be a sequence of random graphs, where $G_n = (V_n, E_n)$ is a graph
with $|V_n|=n$ vertices, $n=n_0,n_0+1,...$ In state $G_n = (V_n, E_n)$, a
vertex $v\in V_n$ is chosen from $V_n$ uniformly at random and is partially
duplicated. Upon such an event, a new vertex $v'\notin V_n$ is created and
every edge $\{v,w\} \in E_n$ is copied with probability~$p$, i.e.\ $E_{n+1}$
has an edge $\{v',w\}$ with probability~$p$, independently of all other edges.
Within this graph, we study several aspects for large~$n$. (i) The frequency of
isolated vertices converges to~1 if $p\leq p^* \approx 0.567143$, the unique
solution of $pe^p=1$. (ii) The number $C_k$ of $k$-cliques behaves like
$n^{kp^{k-1}}$ in the sense that $n^{-kp^{k-1}}C_k$ converges against a
non-trivial limit, if the starting graph has at least one $k$-clique. In
particular, the average degree of a vertex (which equals the number of edges --
or 2-cliques -- divided by the size of the graph) converges to $0$ iff $p<0.5$
and we obtain that the transitivity ratio of the random graph is of the order
$n^{-2p(1-p)}$. (iii) The evolution of the degrees of the vertices in the
initial graph can be described explicitly. Here, we obtain the full
distribution as well as convergence results.
