Random walks on networks with stochastic reset to multiple nodes

In the previous chapters, we explored the effects of resetting on networks considering one and two nodes. In this chapter, we will describe a generalization of random walks with resetting to an arbitrary number of nodes $\mathcal{M}$. In order to make the equations clear and understandable, it is necessary to introduce a more compact notation. Once the theory is fully explained and implemented, we will apply this generalization to more complex structures than the simple ring, particularly to Cayley trees, random distribution of points in a continuous space, and interacting cycles. For this last type of networks, we apply the Google search strategy where the dynamics resets to all nodes. We introduce the total resetting probability $\beta$ and the global mean first passage time $\mathcal{T}$, which is the average of the MFPT over all the target and source nodes, consequently its value does not depend on the initial condition of the random walker. The main objective is to show how the parameter $\beta$ affects $\mathcal{T}$ and in some cases optimizes it.