We introduce a minimal generative model for densifying networks in which a new node attaches to a randomly selected target node and also to each of its neighbors with probability $p$. The networks that emerge from this copying mechanism are sparse for $p<\frac{1}{2}$ and dense (average degree increasing with number of nodes $N$) for $p\ge \frac{1}{2}$. The behavior in the dense regime is especially rich; for example, individual network realizations that are built by copying 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 $N$ dependences 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—all nodes are connected—is nonzero as $N\to \infty$.

• Received 14 July 2016

DOI:https://doi.org/10.1103/PhysRevLett.117.218301