Random walks on complex networks with first-passage resetting

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 walker is reset instantaneously to given node whenever it hits either nodes. derive exact expressions stationary occupation probability, average number resets in long time, mean time between two nonobservable show that all quantities can be expressed terms fundamental matrix $\mathbf{Z}={(\mathbf{I}\ensuremath{-}\mathbf{Q})}^{\ensuremath{-}1}$, where $\mathbf{I}$ identity $\mathbf{Q}$ transition Finally, we use ring networks, two-dimensional square lattices, barbell Cayley trees demonstrate advantage global search such networks.