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

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. The projections never exhibit a negative degree correlation. Positive degree correlations in the networks 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 determined the relation between the finite components in a supercritical phase possessing the giant component and the projection in a subcritical phase when the degree distributions for group and individual are Poisson.