The growing availability of network data and of scientific interest in
distributed systems has led to the rapid development of statistical models of
network structure. Typically, however, these are models for the entire network,
while the data consists only of a sampled sub-network. Parameters for the whole
network, which is what is of interest, are estimated by applying the model to
the sub-network. This assumes that the model is consistent under sampling, or,
in terms of the theory of stochastic processes, that it defines a projective
family.

High density clusters can be characterized by the connected components of a
level set $L(\lambda) = \{x:\ p(x)>\lambda\}$ of the underlying probability
density function $p$ generating the data, at some appropriate level
$\lambda\geq 0$. The complete hierarchical clustering can be characterized by a
cluster tree ${\cal T}= \bigcup_{\lambda} L(\lambda)$. In this paper, we study
the behavior of a density level set estimate $\widehat L(\lambda)$ and cluster
tree estimate $\widehat{\cal{T}}$ based on a kernel density estimator with
kernel bandwidth $h$.

The p_1 model is a directed random graph model used to describe dyadic
interactions in a social network in terms of effects due to differential
attraction (popularity) and expansiveness, as well as an additional effect due
to reciprocation. In this article we carry out an algebraic statistics analysis
of this model. We show that the p_1 model is a toric model specified by a
multi-homogeneous ideal. We conduct an extensive study of the Markov bases for
p_1 models that incorporate explicitly the constraint arising from
multi-homogeneity.