Large-scale behavior of the partial duplication random graph
  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. 
Felix Hermann, Peter Pfaffelhuber
math.PR