  We study discrete-time random walks on networks subject to a time-dependent
stochastic resetting, where the walker either hops randomly between neighboring
nodes with a probability $1-\phi(a)$, or is reset to a given node with a
complementary probability $\phi(a)$. The resetting probability $\phi(a)$
depends on the time $a$ since the last reset event (also called $a$ the age of
the walker). Using the renewal approach and spectral decomposition of
transition matrix, we formulize the stationary occupation probability of the
walker at each node and the mean first passage time between arbitrary two
nodes. Concretely, we consider that two different time-dependent resetting
protocols that are both exactly solvable. One is that $\phi(a)$ is a
step-shaped function of $a$ and the other one is that $\phi(a)$ is a rational
function of $a$. We demonstrate the theoretical results on two different
networks, also validated by numerical simulations, and find that the
time-modulated resetting protocols can be more advantageous than the
constant-probability resetting in accelerating the completion of a target
search process.
