Protein-protein interaction (PPI) networks are commonly explored for the identification of distinctive biological traits, such as pathways, modules, and functional motifs. In this respect, understanding the underlying network structure is vital to assess the significance of any discovered features. We recently demonstrated that PPI networks show degree-weighted behavior, whereby the probability of interaction between two proteins is generally proportional to the product of their numbers of interacting partners or degrees. It was surmised that degree-weighted behavior is a characteristic of randomness. We expand upon these findings by developing a random, degree-weighted, network model and show that eight PPI networks determined from single high-throughput (HT) experiments have global and local properties that are consistent with this model. The apparent random connectivity in HT PPI networks is counter-intuitive with respect to their observed degree distributions; however, we resolve this discrepancy by introducing a non-network-based model for the evolution of protein degrees or "binding affinities." This mechanism is based on duplication and random mutation, for which the degree distribution converges to a steady state that is identical to one obtained by averaging over the eight HT PPI networks. The results imply that the degrees and connectivities incorporated in HT PPI networks are characteristic of unbiased interactions between proteins that have varying individual binding affinities. These findings corroborate the observation that curated and high-confidence PPI networks are distinct from HT PPI networks and not consistent with a random connectivity. These results provide an avenue to discern indiscriminate organizations in biological networks and suggest caution in the analysis of curated and high-confidence networks.