Network structures, such as polycrystals, foams, and biological tissues consist of space-filling polyhedral grains, bubbles, or cells, respectively. The irregularity of such physical polyhedra derives from the fact that their faces consist of mixed shapes, their edges are of varying length, and the positions of their vertices are arranged non-symmetrically. The geometric complexity of irregular polyhedra forming 3-d networks makes their analysis difficult. To help circumvent this difficulty, average N-hedra (ANHs) are proposed as a set of regular polyhedra consisting of N identical faces. ANHsonly a few of which are constructibleact as abstract proxies for each corresponding topological class of irregular network polyhedra with the same number of faces. This study provides a comparison of the intrinsic areas and volumes of ANHs with data estimated numerically using Brakkes Surface Evolver simulations for a range of constructible polyhedral cells. Evolver data show that for every topological class, ANHs always provide a sharp upper bound for the isoperimetric quotient, which is a measure of the inverse energy cost of the cell. Of special interest also is demonstrating that the critical ANH, which has both zero mean and Gaussian curvature, satisfies Kusners bound for the average number of faces in a minimally partitioned network. In Euclidean 3-space the requisite number of faces is satisfied exactly by the critical ANH. The critical ANH, therefore, statistically represents the abstract unit cell that solves the so-called Kelvin Problem for a space-filling 3-d network exhibiting the minimum partition energy or surface area. This limit remains of particular interest in the case of annealed polycrystals and dry foams, as it establishes the lower bound of the energy cost of cellular random structures with a given metric gauge.