  The functions of complex networks are usually determined by a small set of
vital nodes. Finding the best set of vital nodes (eigenshield nodes) is
critical to the network's robustness against rumor spreading and cascading
failures, which makes it one of the fundamental problems in network science.
The problem is challenging as it requires to maximize the influence of nodes in
the set while simultaneously minimizing the redundancies between the set's
nodes. However, the redundancy mechanism is rarely investigated by previous
studies. Here we introduce the matrix perturbation framework to find a small
``eigenshield" set of nodes that, when removed, lead to the largest drop in the
network's spectral radius. We show that finding the ``eigenshield" nodes can be
translated into the optimization of an objective function that simultaneously
accounts for the individual influence of each node and redundancy between
different nodes.
  We analytically quantify the influence redundancy that explains why an
important node might play an insignificant role in the ``eigenshield" node set.
Extensive experiments under diverse influence maximization problems, ranging
from network dismantling to spreading maximization, demonstrate that the
eigenshield detection tends to significantly outperforms state-of-the-art
methods across most problems. Our findings shed light on the mechanisms that
may lie at the core of the function of vital nodes in complex network.
