We consider configuration graphs with vertex degrees distributed independently according to the power law, with a truncated parameter τ normally distributed on the interval (a, b). By computer simulation we study the robustness of such graphs to both random and targeted destructions in the cases of preserving graph connectivity and «forest fire». In the first case, the models of the dependence of the size of the giant component and the probability of graph destruction on the initial graph size N and the percentage of removed vertices were obtained. In the case of the forest fire, the main interest was in the number of vertices remaining after the fire. Along with the two «fire starts» (random ignition and targeted lightning-up of the node with the highest degree), we considered the two fire propagation processes: with a fixed probability of fire transition through a graph edge and a random one.For both types of external destructive impacts, we compared the new results with the ones obtained previously for the configuration graphs with a fixed parameter of node degree distribution and the one uniformly distributed on the interval [a, b].