We consider a discrete-time Markovian random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process. Some nodes of the network are target nodes, and we focus on the statistics of first hitting of these nodes. In the non-Markov case of the renewal process, we consider both light-and fat-tailed inter-reset distributions. We derive the propagator matrix in terms of discrete backward recurrence time PDFs and in the light-tailed case we show the existence of a non-equilibrium steady state. In order to tackle the non-Markov scenario, we derive a defective propagator matrix which describes an auxiliary walk characterized by killing the walker as soon as it hits target nodes. This propagator provides the information on the mean first passage statistics to the target nodes. We establish sufficient conditions for ergodicity of the walk under resetting. Furthermore, we discuss a generic resetting mechanism for which the walk is non-ergodic. Finally, we analyze inter-reset time distributions with infinite mean where we focus on the Sibuya case. We apply these results to study the mean first passage times for Markovian and non-Markovian (Sibuya) renewal resetting protocols in realizations of Watts-Strogatz and Barabási-Albert random graphs. We show non trivial behavior of the dependence of the mean first passage time on the proportions of the relocation nodes, target nodes and of the resetting rates. It turns out that, in the large-world case of the Watts-Strogatz graph, the efficiency of a random searcher particularly benefits from the presence of resets.

Dynamics with stochastic resetting (SR) occurs whenever the time evolution of a phenomenon is characterized by repeated relocations that happen randomly in time, according to a certain mechanism. SR is ubiquitous in nature and society: in foraging, an animal undertakes repeated excursions from its lair to search for food; in biochemistry, when certain biomolecules such as proteins are searching for binding sites; problem-solving strategies; financial markets recurrently hit by crises. Resets may represent catastrophic events such as earthquakes, volcanic eruptions or wood fires after which flora and fauna restart to develop, or seasonal hurricanes forcing human societies to reconstruct infrastructures. The first three mentioned examples fall into the category of random search with resetting. In this respect, a major issue is whether random search strategies are improved when combining them with resetting. In many cases, resetting can indeed significantly reduce the time for reaching a target and it has become a matter of fundamental interest to model such resetting strategies.

In this paper we consider the behavior of a discrete-time Markov walker moving randomly on discrete structures, where we focus in particular on undirected connected graphs. This random motion is subjected to successive recurrent resets (instantaneous relocations to nodes) governed by a counting process which can be of Markovian or non-Markovian nature. We analyze various resetting mechanisms and characterize these choices by means of a relocation matrix. We explore the resulting dynamics on realizations of the Watts-Strogatz and Barabási-Albert random graphs and investigate how resets can affect the efficiency in the searching of a target. We derive an exact formula of the propagator of the associated walk where it turns out that if the waiting time between consecutive resets has a finite mean, the infinite time limit of the propagator matrix is a non-equilibrium steady state (NESS).

Interesting features of the dynamics are revealed by the hitting time statistics for the reaching of target nodes and asymptotic behaviors. In particular, we analyze the mean first passage time for non-Markovian resetting, where we focus on the case of (fat-tailed) Sibuya distributed inter-resetting times.