Within the field of national security and counterterrorism a great need exists to understand covert organizations. To better understand these cellular structures we model and analyze these cells as a collection of subsets of all participants in the covert organization, i.e., as hypergraphs or affiliation networks. Such a covert affiliation network structure is analyzed by evaluating the one-mode projection of the corresponding hypergraph. First we provide a characterization of the total distance in the one-mode projection using its corresponding cell-shrunken version. Secondly we evaluate the one-mode projection with respect to the secrecy versus information tradeoff dilemma every covert organization has to solve. We present and analyze affiliation networks representing common covert organizational forms: star, path and semi-complete hypergraphs. In addition we evaluate an example of a covert organization wishing to conduct an attack and compare its performance to that of the common covert organizational forms. Finally we investigate affiliation networks that are optimal in the sense of balancing secrecy and information. We show how any affiliation tree can be improved by altering its structure. Finally we prove that among covert organizational forms in the class of hypertrees with the same number of cells uniform star affiliation networks are optimal.