  A probability model exhibits instability if small changes in a data outcome
result in large, and often unanticipated, changes in probability. This
instability is a property of the probability model, given by a distributional
form and a given configuration of parameters. For correlated data structures
found in several application areas, there is increasing interest in identifying
such sensitivity in model probability structure. We consider the problem of
quantifying instability for general probability models defined on sequences of
observations, where each sequence of length N has a finite number of possible
values that can be taken at each point. A sequence of probability models
results, indexed by N, and an associated parameter sequence, that accommodates
data of expanding dimension. Model instability is formally shown to occur when
a certain log-probability ratio under such models grows faster than N. In this
case, a one component change in the data sequence can shift probability by
orders of magnitude. Also, as instability becomes more extreme, the resulting
probability models are shown to tend to degeneracy, placing all their
probability on potentially small portions of the sample space. These results on
instability apply to large classes of models commonly used in random graphs,
network analysis, and machine learning contexts.
