  Graph datasets are frequently constructed by a projection of a bipartite
graph, where two nodes are connected in the projection if they share a common
neighbor in the bipartite graph; for example, a coauthorship graph is a
projection of an author-publication bipartite graph. Analyzing the structure of
the projected graph is common, but we do not have a good understanding of the
consequences of the projection on such analyses. Here, we propose and analyze a
random graph model to study what properties we can expect from the projection
step. Our model is based on a Chung-Lu random graph for constructing the
bipartite representation, which enables us to rigorously analyze the projected
graph. We show that common network properties such as sparsity, heavy-tailed
degree distributions, local clustering at nodes, the inverse relationship
between node degree, and global transitivity can be explained and analyzed
through this simple model. We also develop a fast sampling algorithm for our
model, which we show is provably optimal for certain input distributions.
Numerical simulations where model parameters come from real-world datasets show
that much of the clustering behavior in some datasets can just be explained by
the projection step.
