  Social networks and other large sparse data sets pose significant challenges
for statistical inference, as many standard statistical methods for testing
model fit are not applicable in such settings. Algebraic statistics offers a
theoretically justified approach to goodness-of-fit testing that relies on the
theory of Markov bases and is intimately connected with the geometry of the
model as described by its fibers.
  Most current practices require the computation of the entire basis, which is
infeasible in many practical settings. We present a dynamic approach to explore
the fiber of a model, which bypasses this issue, and is based on the
combinatorics of hypergraphs arising from the toric algebra structure of
log-linear models.
  We demonstrate the approach on the Holland-Leinhardt $p_1$ model for random
directed graphs that allows for reciprocated edges.
