  There exist various types of network block models such as the Stochastic
Block Model (SBM), the Degree Corrected Block Model (DCBM), and the Popularity
Adjusted Block Model (PABM). While this leads to a variety of choices, the
block models do not have a nested structure. In addition, there is a
substantial jump in the number of parameters from the DCBM to the PABM. The
objective of this paper is formulation of a hierarchy of block model which does
not rely on arbitrary identifiability conditions. We propose a Nested Block
Model (NBM) that treats the SBM, the DCBM and the PABM as its particular cases
with specific parameter values, and, in addition, allows a multitude of
versions that are more complicated than DCBM but have fewer unknown parameters
than the PABM. The latter allows one to carry out clustering and estimation
without preliminary testing, to see which block model is really true.
