A fundamental issue in the dynamics of complex systems is the resilience of the network in response to targeted attacks. This paper explores the local dynamics of the network attack process by investigating the order of removal of the nodes that have maximal degree, and shows that this dynamic network response can be predicted from the graph's initial connectivity. We demonstrate numerically that the maximal degree M(τ) of the network at time step τ decays exponentially with τ via a topology-dependent exponent. Moreover, the order in which sites are removed can be approximated by considering the network's "hierarchy" function h, which measures for each node Vi how many of its initial nearest neighbors have lower degree versus those that have a higher one. Finally, we show that the exponents we identified for the attack dynamics are related to the exponential behavior of spreading activation dynamics. The results suggest that the function h, which has both local and global properties, is a novel nodal measurement for network dynamics and structure.