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We consider a conditionally Poissonian random graph model where the meandegrees, `capacities', follow a power-tailed distribution with finite mean andinfinite variance. Such a graph of size $N$ has a giant component which issuper-small in the sense that the typical distance between vertices is of theorder of $\log\log N$. The shortest paths travel through a core consisting ofnodes with high mean degrees. In this paper we derive upper bounds of thetypical distance when an upper part of the core is removed, including the casethat the whole core is removed.

On the robustness of power-law random graphs in the finite mean, infinite variance region