Clustering and percolation on superpositions of Bernoulli random graphs

A simple but powerful network model with (Formula presented.) nodes and (Formula presented.) partly overlapping layers is generated as an overlay of independent random graphs (Formula presented.) with variable sizes and densities. The model is parameterized by a joint distribution (Formula presented.) of layer sizes and densities. When (Formula presented.) grows linearly and (Formula presented.) as (Formula presented.), the model generates sparse random graphs with a rich statistical structure, admitting a nonvanishing clustering coefficient together with a limiting degree distribution and clustering spectrum with tunable power-law exponents. Remarkably, the model admits parameter regimes in which bond percolation exhibits two phase transitions: the first related to the emergence of a giant connected component, and the second to the appearance of gigantic single-layer components.