Properties of the connected components in projections of random bipartite networks: effects of clique size fluctuations

Projection is a helpful description for treating bipartite networks as (monopartite) networks with pairwise interactions. Projections induce correlation spontaneously, avoiding negative degree correlation, even if bipartite networks are entirely random. In this study, we examined the structure of projections of random bipartite networks characterized by the degree distribution of individual and group nodes through the generating function method. We decomposed a projection into two subgraphs, the giant component, and finite components and analyzed their degree correlation. We demonstrate that positive degree correlations in projections originating from the clique size fluctuation remain after the decomposition at the set of finite components, although the values of their clustering coefficient are still finite. The giant component can exhibit either positive or negative degree correlations based on the structure of the projection. However, they are positively correlated in most cases. In addition, we found that a projection removed the giant component coincides with one in the subcritical phase, i.e., the discrete duality relation, when the degree distributions for group and individual are of Poisson.