Upper Bounds for the Largest Component in Critical Inhomogeneous Random Graphs

We consider the Norros-Reittu random graph NRn(w), where edges are present independently but edge probabilities are moderated by vertex weights, and use probabilistic arguments based on martingales to analyse the order of the maximal component 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 an unusually large maximal cluster, and simplify the arguments needed to derive polynomial upper bounds for the probability of observing an unusually small largest component.