  We study the asymptotics for sparse exponential random graph models where the
parameters may depend on the number of vertices of the graph. We obtain exact
estimates for the mean and variance of the limiting probability distribution
and the limiting log partition function of the edge-(single)-star model. They
are in sharp contrast to the corresponding asymptotics in dense exponential
random graph models. Similar analysis is done for directed sparse exponential
random graph models parametrized by edges and multiple outward stars.
