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 < 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 ≥ 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 → ∞, so that the network is effectively complete as N → ∞.