  We study discrete-time random walks on arbitrary networks with first-passage
resetting processes. To the end, a set of nodes are chosen as observable nodes,
and the walker is reset instantaneously to a given resetting node whenever it
hits either of observable nodes. We derive exact expressions of the stationary
occupation probability, the average number of resets in the long time, and the
mean first-passage time between arbitrary two non-observable nodes. We show
that all the quantities can be expressed in terms of the fundamental matrix
$\textbf{Z}=(\textbf{I}-\textbf{Q})^{-1}$, where $\textbf{I}$ is the identity
matrix and $\textbf{Q}$ is the transition matrix between non-observable nodes.
Finally, we use ring networks, 2d square lattices, barbell networks, and Cayley
trees to demonstrate the advantage of first-passage resetting in global search
on such networks.
