  In the present work, we study random walks on complex networks subject to
stochastic resetting when the resetting probability is node-dependent. Using a
renewal approach, we derive the exact expressions of the stationary occupation
probabilities of the walker on each node and the mean first passage time
between arbitrary two nodes. Finally, we demonstrate our theoretical results on
three networks with two different resetting protocols, validated by numerical
simulations as well. We find that under a delicate setting it is advantageous
to optimize the efficiency of a global search on such networks by the
node-dependent resetting probability.
