  This manuscript presents a general recursive formula to estimate the size of
fibers associated with algebraic maps from graphs to summary statistics of
importance for social network analysis, such as number of edges (graph
density), degree sequence, degree distribution, mixing by nodal covariates, and
degree mixing. That is, the formula estimates the number of labeled graphs that
have given values for network properties. The proposed approach can be extended
to additional network properties (e.g., clustering) as well as properties of
bipartite networks. For special settings in which alternative formulas exist,
simulation studies demonstrate the validity of the proposed approach. We
illustrate the approach for estimating the size of fibers associated with the
Barab\'{a}si--Albert model for the properties of degree distribution and degree
mixing. In addition, we demonstrate how the approach can be used to assess the
diversity of graphs within a fiber.
