  How the giant component of a network disappears under attacking nodes or
links addresses a key aspect of network robustness, which can be framed into
percolation problems. Various strategies to select the node to be deactivated
have been studied in the literature; for instance, a simple random failure or
high-degree adaptive (HDA) percolation. Recently a new attack strategy based on
a quantity called collective-influence (CI) has been proposed from the
perspective of optimal percolation. By successively deactivating the node
having the largest CI-centrality value, it was shown to be able to dismantle a
network more quickly and abruptly than many of the existing methods. In this
paper, we focus on the critical behaviors of the percolation processes
following degree-based attack and CI-based attack on random networks. Through
extensive Monte Carlo simulations assisted by numerical solutions, we estimate
various critical exponents of the HDA percolation and those of the CI
percolations. Our results show that these attack-type percolation processes,
despite displaying apparently more abrupt collapse, nevertheless exhibit
standard mean-field critical behaviors at the percolation transition point. We
further discover an extensive degeneracy in top-centrality nodes in both
processes, which may provide a hint for understanding the observed results
