  We introduce an evolving network model in which a new node attaches to a
randomly selected target node and also to each of its neighbors with
probability $p$. The resulting network is sparse for $p<\frac{1}{2}$ and dense
(average degree increasing with number of nodes $N$) for $p\geq \frac{1}{2}$.
In the dense regime, individual networks realizations built by this copying
mechanism are disparate and not self-averaging. Further, there is an infinite
sequence of structural anomalies at $p=\frac{2}{3}$, $\frac{3}{4}$,
$\frac{4}{5}$, etc., where the dependences on $N$ of the number of triangles
(3-cliques), 4-cliques, undergo phase transitions. When linking to second
neighbors of the target can occur, the probability that the resulting graph is
complete---where all nodes are connected---is non-zero as $N\to\infty$.
