  We introduce a growing network model---the copying model---in which a new
node attaches to a randomly selected target node and, in addition,
independently to each of the neighbors of the target with copying probability
$p$. When $p<\frac{1}{2}$, this algorithm generates sparse networks, in which
the average node degree is finite. A power-law degree distribution also arises,
with a non-universal exponent whose value is determined by a transcendental
equation in $p$. In the sparse regime, the network is "normal", e.g., the
relative fluctuations in the number of links are asymptotically negligible. For
$p\geq \frac{1}{2}$, the emergent networks are dense (the average degree
increases with the number of nodes $N$) and they exhibit intriguing structural
behaviors. In particular, the $N$-dependence of the number of $m$-cliques
(complete subgraphs of $m$ nodes) undergoes $m-1$ transitions from normal to
progressively more anomalous behavior at a $m$-dependent critical values of
$p$. Different realizations of the network, which start from the same initial
state, exhibit macroscopic fluctuations in the thermodynamic limit---absence of
self averaging. When linking to second neighbors of the target node can occur,
the number of links asymptotically grows as $N^2$ as $N\to\infty$, so that the
network is effectively complete as $N\to \infty$.
