  We consider the Norros-Reittu random graph $NR_n(\textbf{w})$, where edges
are present independently but edge probabilities are moderated by vertex
weights, and use probabilistic arguments based on martingales to analyse the
component sizes in this model when considered at criticality. In particular, we
obtain stronger upper bounds (with respect to those available in the
literature) for the probability of observing unusually large maximal clusters,
and simplify the arguments needed to derive polynomial upper bounds for the
probability of observing unusually small largest components.
