  We study random walks with stochastic resetting to the initial position on
arbitrary networks. We obtain the stationary probability distribution as well
as the mean and global first passage times, which allow us to characterize the
effect of resetting on the capacity of a random walker to reach a particular
target or to explore a finite network. We apply the results to rings, Cayley
trees, random and complex networks. Our formalism holds for undirected networks
and can be implemented from the spectral properties of the random walk without
resetting, providing a tool to analyze the search efficiency in different
structures with the small-world property or communities. In this way, we extend
the study of resetting processes to the domain of networks.
