Structural Transitions in Densifying Networks.

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<1/2 and dense (average degree increasing with number of nodes N) for p≥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=2/3, 3/4, 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→∞.